Example Question - right-angled triangle
Here are examples of questions we've helped users solve.
Demonstration of the Pythagorean Theorem
The image shows an equation representing the Pythagorean Theorem which states that in a right-angled triangle, the square of the length of the hypotenuse \((c)\) is equal to the sum of the squares of the lengths of the other two sides \((a)\) and \((b)\). The Pythagorean Theorem is expressed as: \[ a^2 + b^2 = c^2 \]
Solving for the Length of a Side in a Right-Angled Triangle
The image shows a right-angled triangle, which means we can use the Pythagorean theorem to solve for the missing side, labeled as "x." The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In the given triangle, we have one side that is 9 cm and another side that is 12 cm. Assuming that the 12 cm side is the hypotenuse (since it's opposite what is presumably the right angle), we can compute the length of side "x" using the following equation: \( c^2 = a^2 + b^2 \) Here, \( c \) represents the hypotenuse, while \( a \) and \( b \) represent the other two sides. Rearranging the equation to solve for \( a \), we get: \( a^2 = c^2 - b^2 \) Let's substitute the given values: \( x^2 = 12^2 - 9^2 \) \( x^2 = 144 - 81 \) \( x^2 = 63 \) Taking the square root of both sides to solve for \( x \), we find: \( x = \sqrt{63} \) The square root of 63 can be simplified further by factoring out perfect squares. \( 63 = 9 \times 7 \), and since \( 9 \) is a perfect square, we get: \( x = \sqrt{9 \times 7} \) \( x = \sqrt{9} \times \sqrt{7} \) \( x = 3\sqrt{7} \) Therefore, the length of side \( x \) is \( 3\sqrt{7} \) cm, which cannot be simplified further without a decimal approximation. If a decimal approximation is needed, \( \sqrt{7} \approx 2.645 \), and multiplying this by 3 gives: \( x \approx 3 \times 2.645 \) \( x \approx 7.935 \) So the side "x" is approximately 7.935 cm long.
Calculating the Missing Side Length in a Right-Angled Triangle
The image shows a right-angled triangle with one of the sides adjacent to the right angle labeled "24 cm" and the hypotenuse labeled "26 cm." To find the length of the other side adjacent to the right angle (often referred to as the opposite side or the height in the context of triangles), we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem is usually written as: \[ c^2 = a^2 + b^2 \] where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides. Given that \( c = 26 \) cm and \( b = 24 \) cm, we can solve for \( a \): \[ 26^2 = a^2 + 24^2 \] \[ 676 = a^2 + 576 \] \[ a^2 = 676 - 576 \] \[ a^2 = 100 \] \[ a = \sqrt{100} \] \[ a = 10 \] Therefore, the length of the other side adjacent to the right angle (the opposite side) is 10 cm.
Calculating the Length of the Hypotenuse using the Pythagorean Theorem
Claro, podemos resolver el problema utilizando el Teorema de Pitágoras. El Teorema de Pitágoras establece que en un triángulo rectángulo, el cuadrado de la longitud de la hipotenusa (el lado opuesto al ángulo recto) es igual a la suma de los cuadrados de las longitudes de los catetos (los otros dos lados). La fórmula es: a² + b² = c², donde: - c es la longitud de la hipotenusa, - a y b son las longitudes de los otros dos lados. En la imagen proporcionada, el triángulo rectángulo tiene un lado de longitud 5 y otro de longitud 7. Queremos encontrar la longitud de la hipotenusa (que está marcada con "?"). Aplicando el Teorema de Pitágoras, tenemos: 5² + 7² = c² 25 + 49 = c² 74 = c² Para encontrar la longitud de la hipotenusa (c), tomamos la raíz cuadrada de ambos lados de la ecuación: c = √74 Por lo tanto, la longitud de la hipotenusa es la raíz cuadrada de 74, que es aproximadamente 8.6.
Calculating the Length of a Right-Angled Triangle's Leg
The image shows a right-angled triangle with the lengths of the hypotenuse and one leg provided. The hypotenuse is 75 cm, and one leg, which is opposite the right angle, is 72 cm. We are asked to find the length of the other leg, which I will call "b." To solve for leg "b," we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two legs (a and b). This can be written as: \[ c^2 = a^2 + b^2 \] Where: c = length of the hypotenuse (75 cm) a = length of one leg (72 cm) b = length of the missing leg So, we can rearrange the equation to solve for "b": \[ b^2 = c^2 - a^2 \] Input the known values and solve for "b": \[ b^2 = (75)^2 - (72)^2 \] \[ b^2 = 5625 - 5184 \] \[ b^2 = 441 \] Take the square root of both sides to find the length of "b": \[ b = \sqrt{441} \] \[ b = 21 \] Therefore, the length of the missing leg "b" is 21 cm.
Solving for Missing Leg Length in a Right-Angled Triangle
This image shows a right-angled triangle with one leg of length 75 cm, the hypotenuse of length 72 cm, and the length of the other leg labeled "b." In a right-angled triangle, the Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as: c² = a² + b² where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. However, from the diagram, we see that they've incorrectly labeled the hypotenuse as shorter than one of the legs, which cannot be the case. So, let's assume that 72 cm is actually the other leg, and "c" represents the hypotenuse of 75 cm. Then we have: c² = a² + b² Solving for b, the equation becomes: b² = c² - a² Substitute the given values: b² = 75² - 72² b² = 5625 - 5184 b² = 441 Next, find the square root of both sides to solve for b: b = √441 b = 21 Therefore, the length of the missing leg b is 21 centimeters.
Calculating the Length of the Hypotenuse of a Right-Angled Triangle
The image displays a right-angled triangle with the two legs measuring 60 meters and 80 meters, and the length of the hypotenuse labeled as "c." To find the length of the hypotenuse, we use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The Pythagorean theorem is expressed as: \[ c^2 = a^2 + b^2 \] where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides. Given the side lengths of 60 meters and 80 meters, we can plug them into the equation: \[ c^2 = 60^2 + 80^2 \] \[ c^2 = 3600 + 6400 \] \[ c^2 = 10000 \] To find \( c \), we take the square root of both sides of the equation: \[ \sqrt{c^2} = \sqrt{10000} \] \[ c = 100 \] Thus, the length of the hypotenuse is 100 meters. There is no need to round to the nearest tenth, since we have an exact value.
Calculation of Hypotenuse in a Right-Angled Triangle
The image portrays a right-angled triangle with one leg measuring 60 meters and the other leg measuring 80 meters. To find the length of the hypotenuse (`c`), we would use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (`c`) is equal to the sum of the squares of the lengths of the other two sides (the legs). The formula is: \[ c^2 = a^2 + b^2 \] where `a` and `b` are the lengths of the legs, and `c` is the length of the hypotenuse. Here, `a = 60 m` and `b = 80 m`, so we plug these values into the formula to get: \[ c^2 = (60)^2 + (80)^2 \] \[ c^2 = 3600 + 6400 \] \[ c^2 = 10000 \] To find `c`, we take the square root of both sides: \[ c = \sqrt{10000} \] \[ c = 100 \] Therefore, the length of the hypotenuse, `c`, is 100 meters. There is no need to round since the result is a whole number.
Incomplete Right-Angled Triangle Problem
The image contains a right-angled triangle where the right angle is at vertex B, and the angles at vertices A and C are not specified. The triangle sides opposite to vertices A, B, and C are labeled as 'c', 'a', and 'b' respectively. To help solve the problem, I would need to know what specifically the question is asking about the triangle – for example, to find the lengths of sides, the measure of angles, or to use the Pythagorean theorem. The image is cut off and does not provide the complete question. If you can provide more details or the full question, I'd be happy to assist you further.
Trigonometric Ratios in a Right-Angled Triangle
The image shows a right-angled triangle, ΔABC, with angle A = 60°, angle B = 90°, and the hypotenuse AC = 20 ft. We are required to solve a series of questions related to this triangle. a) Using trigonometric ratios to find the perpendicular (opposite side to angle A, which is BC) and base (adjacent side to angle A, which is AB): We use the sine and cosine functions, which are defined as follows for a right-angled triangle: - Sine of an angle is the ratio of the length of the opposite side to the hypotenuse. - Cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. For angle A, sin(60°) = opposite/hypotenuse => sin(60°) = BC/AC BC = AC * sin(60°) BC = 20 * (√3/2) (since sin(60°) is √3/2) BC = 10√3 ft For the base (AB), cos(60°) = adjacent/hypotenuse => cos(60°) = AB/AC AB = AC * cos(60°) AB = 20 * (1/2) (since cos(60°) is 1/2) AB = 10 ft b) Using the perpendicular (BC) and base (AB) obtained above, find the value of tan(60°): Tangent of an angle is the ratio of the length of the opposite side to the adjacent side. tan(60°) = opposite/adjacent tan(60°) = BC/AB tan(60°) = (10√3)/10 tan(60°) = √3 c) Find the values of Sin C and Cos C: In a right triangle, the sine of one non-right angle is the cosine of the other, and vice versa. Since angle C is the 90-angle B (90° - 60° = 30°), we can find the sine and cosine of angle C by using their known values at 30°. sin(C) = sin(30°) = 1/2 cos(C) = cos(30°) = √3/2 d) Using the values of Sin C and Cos C, prove that Sin² C + Cos² C = 1: This is a well-known trigonometric identity known as the Pythagorean identity. Now let's substitute the values obtained for Sin C and Cos C: Sin² C + Cos² C = (1/2)² + (√3/2)² Sin² C + Cos² C = 1/4 + 3/4 Sin² C + Cos² C = 4/4 Sin² C + Cos² C = 1 This proves the Pythagorean identity for angle C.
Calculating Hypotenuse Using Pythagorean Theorem
The Pythagorean Theorem states that for a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be expressed as: \[ c^2 = a^2 + b^2 \] In the given triangle, the lengths of the two sides (a and b) are 4 and 8, respectively. Let's use this information to find the length of the hypotenuse (c): \[ c^2 = 4^2 + 8^2 \] \[ c^2 = 16 + 64 \] \[ c^2 = 80 \] Now we'll find the square root of both sides to solve for c: \[ c = \sqrt{80} \] The exact value of √80 can be simplified by factoring out squares from 80: \[ \sqrt{80} = \sqrt{16 \cdot 5} \] \[ \sqrt{80} = \sqrt{16} \cdot \sqrt{5} \] \[ \sqrt{80} = 4\sqrt{5} \] Therefore, the exact value of c is: \[ c = 4\sqrt{5} \]
Calculating the Length of the Hypotenuse in a Right-Angled Triangle
The image shows a right-angled triangle with one side measuring 13 cm, which is the height, and the base measuring 23 cm. To solve for the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Let's denote the hypotenuse as "c". The formula of the Pythagorean theorem is: a² + b² = c² Where: - a = one leg of the triangle (13 cm) - b = the other leg of the triangle (23 cm) - c = the length of the hypotenuse (what we need to find out) So we have: (13 cm)² + (23 cm)² = c² 169 cm² + 529 cm² = c² 698 cm² = c² Now, take the square root of both sides to find the length of the hypotenuse: c = √698 cm² c ≈ √700 cm² c ≈ 26.46 cm Therefore, the length of the hypotenuse is approximately 26.46 cm.
Solving for Missing Leg Length of a Right Triangle
The given image shows a right triangle with one leg measuring 13 cm, and the hypotenuse measuring 23 cm. The question likely asks for the length of the other leg of the triangle. We can use the Pythagorean theorem to solve for the missing length. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). The formula can be written as: \[ c^2 = a^2 + b^2 \] where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides. Using this theorem and substituting the known values, we get: \[ 23^2 = 13^2 + b^2 \] \[ 529 = 169 + b^2 \] \[ b^2 = 529 - 169 \] \[ b^2 = 360 \] \[ b = \sqrt{360} \] \[ b = 19 \] So the length of the missing leg \( b \) is 19 cm.
Calculating the Length of a Hypotenuse
The Pythagorean theorem states that for a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The theorem can be written as: c² = a² + b² In the provided right-angled triangle, side a is 9 ft and side b is 12 ft. You are asked to find the length of the hypotenuse (c). Using the Pythagorean theorem: c² = 9² + 12² c² = 81 + 144 c² = 225 To find the length of c, take the square root of both sides: c = √225 c = 15 ft Therefore, the length of the hypotenuse is 15 feet.
Solving Right-Angled Triangle Problems with the Pythagorean Theorem
The image shows two separate mathematical problems involving right-angled triangles, one labeled as question 4 and the other as question 5. Both problems are about finding the length of the missing side of the triangle, which is a classic Pythagorean theorem problem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. For question 4, the sides of the triangle are given as 3 (one leg) and 4 (the other leg), and we are to find the hypotenuse. By applying the Pythagorean theorem (a^2 + b^2 = c^2), where 'c' is the hypotenuse and 'a' and 'b' are the other sides, we can solve the triangle: 3^2 + 4^2 = c^2 9 + 16 = c^2 25 = c^2 c = √25 c = 5 So, the length of the hypotenuse is 5 and the correct answer is not shown as marked in the image. For question 5, the sides of the triangle are given as 7 (one leg) and 24 (the other leg), and we are to find the hypotenuse. We apply the Pythagorean theorem again: 7^2 + 24^2 = c^2 49 + 576 = c^2 625 = c^2 c = √625 c = 25 The length of the hypotenuse for question 5 is therefore 25, which corresponds with answer choice B. To summarize: - Question 4: The hypotenuse = 5 (not in the choices) - Question 5: The hypotenuse = 25 (choice B)
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