Question

a wall in marias bedroom is in the shape of a trapezoid. the wall can be divided into a rectangle and a triangle. using the 45°-45°-90° triangle theorem, find the value of h, the height of the wall. 6.5 ft 6.5√2 ft 13 ft 13√2 ft

Explanation:

Step1: Recall 45°-45°-90° triangle theorem

In a 45°-45°-90° triangle, the legs are equal, and the hypotenuse \( c \) is related to the leg \( a \) by \( c = a\sqrt{2} \), or \( a=\frac{c}{\sqrt{2}} \). Here, the hypotenuse of the right triangle (the slant side) is \( 13\sqrt{2} \) ft, and \( h \) is one of the legs (since it's a 45°-45°-90° triangle, the height \( h \) and the base of the triangle are equal legs).

Step2: Solve for \( h \)

Using the formula \( c = a\sqrt{2} \), where \( c = 13\sqrt{2} \) and \( a = h \). Substitute \( c \) into the formula:

$$ 13\sqrt{2}=h\sqrt{2} $$

Divide both sides by \( \sqrt{2} \):

$$ h = \frac{13\sqrt{2}}{\sqrt{2}} = 13 $$

Wait, no, wait. Wait, maybe I misread. Wait, actually, let's check again. Wait, the hypotenuse is \( 13\sqrt{2} \), so using \( c = a\sqrt{2} \), so \( a=\frac{c}{\sqrt{2}} \). Wait, no, if the triangle is 45-45-90, then legs are equal, hypotenuse is leg\( \times\sqrt{2} \). So if hypotenuse is \( 13\sqrt{2} \), then leg \( h=\frac{13\sqrt{2}}{\sqrt{2}} = 13 \)? Wait, no, that would be 13, but wait, maybe the hypotenuse is \( 13\sqrt{2} \), so leg is \( 13 \)? Wait, let's do it again. Let \( h \) be the leg, hypotenuse \( c = h\sqrt{2} \). So \( h=\frac{c}{\sqrt{2}} \). Given \( c = 13\sqrt{2} \), then \( h=\frac{13\sqrt{2}}{\sqrt{2}} = 13 \). So \( h = 13 \) ft. Wait, but let's check the options. The options are 6.5, 6.5√2, 13, 13√2. Wait, maybe I made a mistake. Wait, maybe the hypotenuse is 13√2, but maybe the triangle is isoceles right triangle, so legs are equal. Wait, maybe the hypotenuse is 13√2, so leg is 13? Wait, no, wait, 6.5*2=13. Wait, maybe I misread the hypotenuse. Wait, the hypotenuse is 13√2? Wait, no, maybe the hypotenuse is 13√2, but let's recalculate. Wait, if hypotenuse is \( c = a\sqrt{2} \), then \( a = \frac{c}{\sqrt{2}} \). So \( c = 13\sqrt{2} \), so \( a = \frac{13\sqrt{2}}{\sqrt{2}} = 13 \). So \( h = 13 \) ft. So the answer is 13 ft.

Wait, but let's check again. Let's suppose the hypotenuse is \( 13\sqrt{2} \), then leg \( h = \frac{13\sqrt{2}}{\sqrt{2}} = 13 \). Yes, that's correct. So \( h = 13 \) ft.

Answer:

13 ft (corresponding to the option "13 ft")