Paul walks 25 feet away from his house and places a mirror on the ground. he backs 5 feet away from the mirror so that he can see the tip of the roof. paul's eyes are 6 feet above the ground. paul and the house are both perpendicular to the ground. the angles between the top of the house, the mirror, and the ground and between paul's eyes, the mirror, and the ground are congruent as shown in the image below:
image depicts a mirror on the ground between a person and a house. the mirror is 5 feet away from the person and 25 feet away f
what is the height of the house? show your work and explain your reasoning in complete sentences.

The height of the house is 30 feet

Step-by-step explanation:

* Lets explain how to solve the problem

- Name the place of the mirror with M

- Name the place of Paul with P

∴ MP is the distance between Paul and the mirror

∵ The distance between Paul and the mirror is 5 feet

∴ MP = 5

- Name the place of Paul's eyes with E

- Name the tip of the roof with T

∴ MET is a line segment

- Name the place of the house on the ground with H

- The line TH represent the height of the house

∴ TH = the height of the house

- The line HM is the distance between the house and the mirror

∵ The distance between the house and the mirror is 25 feet

∴ HM = 25

- Paul and the house are both perpendicular to the ground

∴ TH and EP ⊥ HM at H and E

∴ TH // EP

- Paul's eyes are 6 feet above the ground

∴ EP = 6

- All these situations represent two triangles THM and EPM

- Δ THM represents the house , the tip of the roof and the mirror

- Δ EPM represents Paul's eyes , Paul position and the mirror

- In Δ THM

∵ TH // EP

∴ m∠ THM = m∠ EPM ⇒ alternate angles

∴ m∠ HTM = m∠ PEM ⇒ alternate angles

∴ Δ THM ≈ Δ EPM ⇒ AA similarity

∴ EP/TH = MP/MH

∵ EP = 6 , MP = 5 , MH = 25

∴ 6/TH = 5/25 ⇒ by using cross multiplication

∴ 5 TH = 6 × 25

∴ 5 TH = 150 ⇒ divide both sides by 5

∴ TH = 30

- TH is the height of the house

* The height of the house is 30 feet

The height of the house is 30 feet.

Explanation: Paul walks 25 feet away from his house. He stops places a mirror and walks back 5 feet again. The height of Paul from the ground to his eyes are 6 feet. From these statements and the picture given, we can see there are 2 similar triangles, so the angles from one triangle must be congruent to the angles of the other triangle. To solve we can use the equation 6/5=x/25. X= height of the house and 6 = height from the ground to paul's eyes and 5 is the backed-up distance from the mirror.

Solving 6/5 = x/25:

25/1 x / 6/5 = x/25 x 25/1

= 5x6=x/1

=30=x

Therefore the height of the house is 30 feet.

step-by-step explanation:

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