Question

jahzaira wanted to know the height of a flag pole in her neighborhood. she stood a total of 15 feet away from the flagpole and placed a mirror in between. jahzaira’s eyes are 1.25m above the ground and the mirror is 1.55m from her. find the height of the flag pole.
7a. what is the height of the flagpole?
7b. describe how you determined your answer. (6 points!)

Explanation:

Step1: Identify Similar Triangles

The two triangles (from Jahzaira to mirror and flagpole to mirror) are similar by the Law of Reflection (angles are equal, right angles exist), so their sides are proportional. Let \( h \) be the flagpole height. The distance from Jahzaira to mirror is \( 1.55 \, \text{m} \), from mirror to flagpole is \( 13.35 \, \text{m} \), Jahzaira's height is \( 1.25 \, \text{m} \).

Step2: Set Up Proportion

Using similar triangles, \(\frac{\text{Jahzaira's height}}{\text{Distance to mirror}} = \frac{\text{Flagpole height}}{\text{Distance to flagpole from mirror}}\)
So, \(\frac{1.25}{1.55} = \frac{h}{13.35}\)

Step3: Solve for \( h \)

Cross - multiply: \( 1.55h = 1.25\times13.35 \)
Calculate \( 1.25\times13.35 = 16.6875 \)
Then \( h=\frac{16.6875}{1.55} = 10.766\approx10.77 \)? Wait, no, wait: Wait, total distance from Jahzaira to flagpole is 15 feet? Wait, no, the units: Wait, the diagram has 1.55 m (Jahzaira to mirror), 13.35 m (mirror to flagpole), so total horizontal distance is \( 1.55 + 13.35=14.9\approx15 \) m (matches the 15 feet? Wait, maybe unit conversion error, but proceed with meters. Wait, no, let's re - check:

Wait, correct proportion: \(\frac{1.25}{1.55}=\frac{h}{13.35 + 1.55}\)? No, no. Wait, the horizontal distance from Jahzaira to mirror is \( 1.55 \, \text{m} \), from mirror to flagpole is \( 13.35 \, \text{m} \). So the two triangles: one with height \( 1.25 \) and base \( 1.55 \), the other with height \( h \) and base \( 13.35 + 1.55 \)? No, no, the diagram: the vertical side for Jahzaira is \( 1.25 \), horizontal to mirror is \( 1.55 \). The flagpole's horizontal distance from mirror is \( 13.35 \), vertical height \( h \). So the two right - triangles: angle at mirror is equal (reflection), so similar. So \(\frac{1.25}{1.55}=\frac{h}{13.35}\)? Wait, no, the distance from Jahzaira to mirror is \( 1.55 \), from mirror to flagpole is \( 13.35 \), so the base of the flagpole's triangle is \( 13.35 \), and Jahzaira's is \( 1.55 \). So:

\(\frac{1.25}{1.55}=\frac{h}{13.35}\)

\( h=\frac{1.25\times13.35}{1.55}=\frac{16.6875}{1.55}=10.766\approx10.77 \)? Wait, that can't be. Wait, maybe I mixed up the distances. Wait, the total distance from Jahzaira to flagpole is 15 feet? Wait, the problem says "a total of 15 feet away from the flagpole and placed a mirror in between". Oh! Units: 15 feet. Let's convert 15 feet to meters: 1 foot = 0.3048 m, 15 feet = \( 15\times0.3048 = 4.572 \) m. Wait, but the diagram has 1.55 m and 13.35 m, which sum to 14.9 m, which is not 4.572 m. There's a unit confusion. Wait, maybe the 15 feet is a typo, and it's 15 meters. Let's proceed with the diagram's meters.

Wait, no, let's re - examine the problem: "She stood a total of 15 feet away from the flagpole and placed a mirror in between. Jahzaira’s eyes are 1.25m above the ground and the mirror is 1.55m from her." Wait, 15 feet is about 4.572 m, but the mirror to flagpole is 13.35 m, which is more than 4.572 m. So there's a unit error. But assuming that the horizontal distances are 1.55 m (Jahzaira to mirror) and \( 15 - 1.55 \) feet? No, this is confusing. Wait, maybe the 15 feet is 15 meters. Let's proceed with the similar triangles:

Let \( d_1 = 1.55 \, \text{m} \) (Jahzaira to mirror), \( d_2=13.35 \, \text{m} \) (mirror to flagpole), \( h_1 = 1.25 \, \text{m} \) (Jahzaira's height), \( h_2 \) (flagpole height).

Since triangles are similar, \(\frac{h_1}{d_1}=\frac{h_2}{d_2}\)

So \( h_2=\frac{h_1\times d_2}{d_1}=\frac{1.25\times13.35}{1.55}=\frac{16.6875}{1.55} = 10.766\approx10.77 \, \text{m}\)? Wait, but that seems low. Wait, maybe the distance from Jahzaira to mirror is 1.55 m, and from Jahzaira to flagpole is 15 m, so mirror to flagpole is \( 15 - 1.55 = 13.45 \) m (close to 13.35, maybe rounding). Let's use 15 m total distance. So \( d_1 = 1.55 \, \text{m} \), \( d_2=15 - 1.55 = 13.45 \, \text{m} \)

Then \( h_2=\frac{1.25\times13.45}{1.55}=\frac{16.8125}{1.55}\approx10.847\approx10.85 \, \text{m}\)

Wait, maybe the correct proportion is \(\frac{\text{Jahzaira's height}}{\text{Distance from Jahzaira to mirror}}=\frac{\text{Flagpole height}}{\text{Distance from mirror to flagpole}}\)

So \( h=\frac{1.25\times13.35}{1.55}=10.77 \, \text{m}\) (approx). But let's check the calculation again:

\( 1.25\times13.35 = 16.6875 \)

\( 16.6875\div1.55 = 10.766\approx10.77 \)

But maybe the problem expects using the 15 feet as 15 meters (since the other distances are in meters). So the flagpole height is approximately \( 10.77 \) meters. Wait, but let's check the similar triangles again. The two triangles are similar, so the ratio of height to horizontal distance is the same.

Answer:

The height of the flagpole is approximately \(\boldsymbol{10.77 \, \text{m}}\) (or if we consider the total distance as 15 feet converted to meters and recalculate, but based on the diagram's meters, the calculation gives approximately \( 10.77 \, \text{m} \)).