Question

15 in a right isosceles triangle, the hypotenuse measures 10 cm. what is the measure of each leg? a 5 cm b 5√2 cm c 10 cm d 10√2 cm
14 this diagram shows a right triangle.
what is the length of side mt? a 5√2 cm b 6 cm c 5√3 cm d 6√3 cm
16 this diagram shows △mnt.
what is the length of side nt? a 6 cm b 6√2 cm c 12 cm d 12√2 cm
19 john leans a 10 - foot ladder against his house. the ladder forms a 30° angle with the ground. what is the approximate distance from the base of the ladder to the base of the house? a 2.9 ft b 3.5 ft c 7.1 ft d 8.7 ft

Explanation:

Step1: Identify the type of triangle

The triangle is a right - isosceles triangle. In a right - isosceles triangle (45 - 45 - 90 triangle), the ratio of the sides is \(1:1:\sqrt{2}\), where the legs are of equal length and the hypotenuse \(c\) is related to the leg \(a\) by the formula \(c = a\sqrt{2}\).

Step2: Use the hypotenuse - leg relationship

Given that the hypotenuse \(c = 10\) cm. Let the length of each leg be \(x\). Using the formula \(c=x\sqrt{2}\), we can solve for \(x\). Rearranging the formula gives \(x=\frac{c}{\sqrt{2}}\). Rationalizing the denominator: \(x = \frac{10}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{10\sqrt{2}}{2}=5\sqrt{2}\) cm.

Answer:

\(5\sqrt{2}\) cm