Example Question - triangle side lengths
Here are examples of questions we've helped users solve.
Solving Triangles with Given Ratios
The image shows two triangles, with a given statement that "The ratio of the LARGER triangle to the SMALLER triangle is \( \frac{3}{5} \)." To use this information to solve the problem, we must understand that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. In this case, the larger triangle has sides of lengths 21 and 18, but we are not given the corresponding sides of the smaller triangle. However, we do know one of the sides of the smaller triangle, which is 5. Since we know the ratio of the areas is \( \frac{3}{5} \), we can set up an equation using the side of the smaller triangle that we know: \[ \left( \frac{\text{Side of larger triangle}}{\text{Side of smaller triangle}} \right)^2 = \frac{3}{5} \] We can use either 21 (corresponding to the larger side) or 18 (corresponding to the smaller side) for the "Side of larger triangle" in the equation. Let's use 18 as it corresponds with the side length of 5 in the smaller triangle: \[ \left( \frac{18}{5} \right)^2 = \frac{3}{5} \] To find the "Side of larger triangle" that corresponds to the smaller triangle's side of length 5, we can use the square root to get: \[ \left( \frac{18}{5} \right) = \sqrt{\frac{3}{5}} \] \[ \frac{18}{5} = \frac{\sqrt{3}}{\sqrt{5}} \] Now, to solve for the corresponding side in the larger triangle, we cross-multiply: \[ 5\sqrt{3} = 18\sqrt{5} \] \[ \text{Side of larger triangle} = \frac{18\sqrt{5}}{\sqrt{3}} \] \[ \text{Side of larger triangle} = \frac{18\sqrt{3}\sqrt{5}}{3} \] \[ \text{Side of larger triangle} = 6\sqrt{15} \] So the side of the larger triangle that corresponds to the side of length 5 in the smaller triangle has a length of \( 6\sqrt{15} \), which would be the correct value to fill in the blank space in the image.
Calculating the Length of a Right Triangle's Hypotenuse
The triangle shown in the image is a right triangle, and we are asked to find the length of the hypotenuse (labeled "c"). To do this, we can use the Pythagorean theorem, which states that in a right 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 formula for the Pythagorean theorem is: a² + b² = c² From the image, we have side lengths: a = 60 meters (the horizontal side) b = 80 meters (the vertical side) Now we substitute these values into the formula: (60 m)² + (80 m)² = c² 3600 m² + 6400 m² = c² 10000 m² = c² Taking the square root of both sides to solve for c, we get: c = √10000 m² c = 100 m Therefore, the length of the hypotenuse c is 100 meters. There is no need to round because the result is an exact whole number.
Solving Triangle Side Segment Lengths with the Angle Bisector Theorem
The image depicts a geometrical problem involving a triangle with vertices labeled as A, B, and C. The sides opposite these vertices are labeled with their respective lengths: side BC measures 29 units, side AC measures 21 units, and side AB measures 20 units. There is also an angle bisector drawn from vertex B to side AC, dividing it into two segments (not labeled in the picture). To solve for the unknown length of the segments into which the angle bisector divides side AC, we can use the Angle Bisector Theorem, which states that the angle bisector of a triangle divides the opposite side into two segments that are proportional to the adjacent sides. Let x be the length of the segment from point A to the point where the bisector intersects side AC, and let (21 - x) be the length of the segment from that point to point C. According to the Angle Bisector Theorem: x / (21 - x) = AB / BC Substituting the given lengths: x / (21 - x) = 20 / 29 Cross-multiplying gives: 29x = 20(21 - x) 29x = 420 - 20x 29x + 20x = 420 49x = 420 x = 420 / 49 x = 8.57 (rounded to two decimal places) So, one segment is approximately 8.57 units long. To find the length of the other segment, we subtract this from the total length of side AC: 21 - 8.57 = 12.43 (rounded to two decimal places) Thus, the angle bisector divides side AC into two segments approximately 8.57 units and 12.43 units in length.
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.
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