Question

part 1 of 3
emaan designs the
birdhouse shown to the
right.
a. what is the total height
of the birdhouse? show
your work.
b. if emaan decides to
change the design by
increasing each side of
the roof from 12.5 cm to
15.2 cm, what will be the
new height of
the birdhouse? all other
labeled dimensions on
the birdhouse will remain
unchanged.
a. what is the equation to use to find the height of the top part
of the birdhouse? choose the correct answer below.

Explanation:

Step1: Analyze the roof triangle

The roof is an isosceles triangle with equal sides (12.5 cm) and base related to the 9.2 cm segments (the base of the triangle is \(9.2 + 9.2 = 18.4\) cm? Wait, no, actually, the two 9.2 cm are the horizontal segments? Wait, looking at the diagram, the vertical line splits the roof triangle into two right triangles. Each right triangle has hypotenuse 12.5 cm and base 9.2 cm (since the horizontal distance from the center to the side is 9.2 cm). So to find the height \(h\) of the roof triangle (the vertical part), we use the Pythagorean theorem: \(h=\sqrt{12.5^{2}-9.2^{2}}\). Wait, but the question is about the equation to find the height of the top part (the roof's height). The top part is a triangle, and we can use the Pythagorean theorem here. Let's denote the height of the roof triangle as \(h\), the hypotenuse as \(c = 12.5\) cm, and the base of the right triangle (half the base of the isosceles triangle) as \(a = 9.2\) cm. Then by Pythagoras, \(h=\sqrt{c^{2}-a^{2}}\), so the equation is \(h = \sqrt{12.5^{2}-9.2^{2}}\) (or rearranged, but the key is using Pythagorean theorem for the right triangle formed by the roof's side, the horizontal segment, and the vertical height of the roof).

Step2: Determine the equation

The top part (roof) is a triangle, and when we split it into two right triangles (by the vertical line), each right triangle has hypotenuse (roof side) = 12.5 cm, one leg (horizontal) = 9.2 cm, and the other leg (vertical height of roof) = \(h\). So by Pythagorean theorem: \(h^{2}+9.2^{2}=12.5^{2}\), so solving for \(h\), we get \(h=\sqrt{12.5^{2}-9.2^{2}}\). So the equation to find the height of the top part is based on the Pythagorean theorem, using the roof side (12.5 cm) as the hypotenuse and the horizontal segment (9.2 cm) as one leg, to find the vertical leg (height of roof).

Answer:

The equation to find the height of the top part (roof) is \(h = \sqrt{12.5^{2}-9.2^{2}}\) (or in the form of the Pythagorean theorem: \(h^{2}+9.2^{2}=12.5^{2}\) where \(h\) is the height of the top part).