Example Question - side lengths
Here are examples of questions we've helped users solve.
Triangle Geometry Problem
<p>Given triangle ABC with angle A = 32° and sides AC = 8 and BC = 5.5, we will use the Law of Sines to find side AB.</p> <p>According to the Law of Sines:</p> <p> \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] </p> <p>Let AB = c, then:</p> <p> \[ \frac{c}{\sin(32°)} = \frac{8}{\sin(B)} \] \end{p> <p>To find angle B, we can use the sine rule again:</p> <p> \[ \frac{5.5}{\sin(B)} = \frac{8}{\sin(32°)} \end{p> <p>Rearranging gives:</p> <p> \[ \sin(B) = \frac{5.5 \cdot \sin(32°)}{8} \end{p> <p>Calculating sin(B) and then angle B, we can find angle C = 180° - A - B.</p> <p>Finally, using angle C, apply the Law of Sines again to find side c:</p> <p> \[ \frac{c}{\sin(C)} = \frac{5.5}{\sin(B)} \end{p>
Triangle Side Relationships
Para resolver el problema, necesitamos usar el teorema de Pitágoras para verificar la validez de las longitudes de los lados del triángulo. El teorema de Pitágoras se expresa como \( c^2 = a^2 + b^2 \), donde \( c \) es la longitud de la hipotenusa y \( a \) y \( b \) son las longitudes de los catetos. <p>Tomamos los valores dados en la imagen:</p> <p>\( a = 9 \) cm (cateto opuesto)</p> <p>\( b = 12 \) cm (cateto adyacente)</p> <p>\( c = 16 \) cm (hipotenusa)</p> <p>Aplicamos el teorema de Pitágoras:</p> \[ c^2 = a^2 + b^2 \] \[ 16^2 = 9^2 + 12^2 \] \[ 256 = 81 + 144 \] \[ 256 = 225 \] <p>Como \( 256 \neq 225 \), la relación dada no cumple con el teorema de Pitágoras. Por lo tanto, las longitudes dadas para los lados del triángulo son incorrectas.</p>
Calculating side lengths of a 30-60-90 triangle
The image shows a right-angled triangle with one angle marked as 30 degrees and the side opposite to this angle is given as 6. In a right-angled triangle where one of the angles is 30 degrees, we can apply the special right triangle ratios that hold true for 30-60-90 triangles. In a 30-60-90 triangle, the ratio of the lengths of the sides opposite the 30°, 60°, and 90° angles is 1:√3:2, respectively. Since the side opposite the 30° angle (the shortest side) is given as 6, we can determine the lengths of the other sides using the ratio: - The length of the hypotenuse (opposite the 90° angle) is twice the length of the side opposite the 30° angle. So, the hypotenuse = 2 × 6 = 12. - The length of the side opposite the 60° angle (the longer leg) is √3 times the length of the side opposite the 30° angle. So, the longer leg = √3 × 6 = 6√3. In summary, the lengths of the sides of the triangle are: - Shortest side (opposite 30°): 6 - Longer leg (opposite 60°): 6√3 - Hypotenuse (opposite 90°): 12 These are the side lengths of the triangle based on the given information.
Calculating Side Lengths of a 30-60-90 Triangle
The image shows a right-angled triangle with one angle of 30 degrees, indicating that this is a 30-60-90 triangle, a special type of right triangle. The side opposite the 30-degree angle, the shortest side, is labeled as 6 units in length. In a 30-60-90 triangle, the lengths of the sides are in a consistent ratio. The side opposite the 30-degree angle (the shortest side) is typically labeled as 'x'. The side opposite the 60-degree angle (the longer leg) is '√3 * x', and the side opposite the 90-degree angle (the hypotenuse) is '2x'. Given that the shortest side is 6 units (x = 6), we can find the lengths of the other two sides as follows: - The longer leg (60-degree side) = √3 * x = √3 * 6 = 6√3 - The hypotenuse (90-degree side) = 2x = 2 * 6 = 12 So, the side opposite the 60-degree angle is 6√3 units long, and the hypotenuse is 12 units long.
Congruent Triangles and Side Lengths
The image shows a pair of congruent triangles, ΔDEG and ΔEFG, with DE congruent to EF, DG equal to 3a, and FG equal to a + 42. In congruent triangles, corresponding sides are equal in length. Therefore: DE = EF Since FG is the sum of DG and EG, and EG is equal to DE (because DE = EF and EF = EG by congruency), you can express FG as: FG = DG + EG Given: DG = 3a EG = DE = EF (Because of the congruency between ΔDEG and ΔEFG) Since DE is congruent to EF, that implies EG = EF. So using the information that FG = DG + EG, we can substitute the given values into the equation: FG = 3a + EF We were also given that FG = a + 42. This allows us to set up the following equation since they both represent FG: 3a + EF = a + 42 However, to find FG, we do not actually need to solve for a or EF individually since FG equals a + 42 by the given information. Therefore: FG = a + 42 This is the expression for FG, and without additional information or numerical values provided for a, this is as simplified as it gets.
Finding Side Lengths of Similar Polygons
The image shows two similar polygons labeled GHJK and RQTS. To find the side lengths RS and TU, we'll use the properties of similar polygons. In similar polygons, the ratios of the corresponding side lengths are equal. From the image, we can see the corresponding side lengths of the polygons as follows: GH (45 units) corresponds to RQ (unknown) HJ (27 units) corresponds to QS (37 units) JK (18 units) corresponds to ST (unknown) KG (36 units) corresponds to TR (52 units) To find the length of RS, we compare the ratios of HJ to QS (which we know) with the ratio of GH to RQ (which we need to find out): HJ/QS = GH/RQ 27/37 = 45/RQ Now, let's solve for RQ (denoted as RS in the question): RQ = (45 * 37) / 27 Calculate the value of RQ: RQ = 1665 / 27 ≈ 61.667 So, RS ≈ 61.7 (rounded to one decimal place). To find the length of TU (which corresponds to JK in the larger polygon), we'll set up a similar ratio. This time, we'll use the sides KG and TR since their lengths are known: JK/ST = KG/TR 18/TU = 36/52 Now solve for TU: TU = (18 * 52) / 36 Calculate the value of TU: TU = 936 / 36 = 26 The length of TU is exactly 26 units. Therefore, RS ≈ 61.7 and TU = 26.
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