Question

(03.06 MC)

The diagram below models the layout at a carnival where G, R, P, C, B, and E are various
locations on the grounds. GRPC is a parallelogram.
400 ft
450 ft
350 ft
B 250 ft
E
P
R
Part A: Identify a pair of similar triangles. (2 points)
Part B: Explain how you know the triangles from Part A are similar. (4 points)
Part C: Find the distance from B to E and from P to E. Show your work. (4 points)

Answer 1

Triangles GBC and BEP are similar by AA similarity. Using proportions with known side lengths, the distance from B to E is approximately 321.42 ft, and the distance from P to E is approximately 90.9 ft.

Part A: Identifying Similar Triangles

The pair of similar triangles in the diagram are triangle GBC and triangle BEP.

Part B: Explaining Similarity

These triangles are similar because they share two pairs of corresponding angles that are congruent.

Angle GBC and angle BEP: Both angles are formed by a perpendicular line intersecting a parallel line. These angles are always congruent.

Angle GCB and angle BPE: These angles are alternate interior angles formed by transversal line PE intersecting parallelograms GRPC and BEP. Alternate interior angles are always congruent.

Therefore, triangles GBC and BEP are similar by the Angle-Angle (AA) similarity criterion.

Part C: Finding Distances

To find the distances from B to E and P to E, we can use the following steps:

1. Set up the proportion: Since triangles GBC and BEP are similar, we can set up a proportion of their corresponding side lengths. Let x be the distance from B to E and y be the distance from P to E. We know the side lengths of triangle GBC from the diagram: GC = 400 ft, CB = 350 ft, and GB = 450 ft. We also know the length of BP from the diagram: BP = 250 ft.

2. Write the proportion: Using the AA similarity criterion, we can write the following proportion:

GB/CB = BE/x

And, using the similarity of alternate interior angles, we can write another proportion:

GC/BP = BE/y

3. Substitute and solve: Since BE appears in both proportions, we can substitute it to eliminate it and solve for x and y. From the first proportion, we can express BE as:

BE = (GB/CB) * x

Substitute this expression for BE in the second proportion:

GC/BP = ((GB/CB) * x) / y

Simplify and solve for x:

y * GC = x * GB / CB

y = (x * GB) / (CB * GC)

y = (x * 450) / (350 * 400)

y ≈ 0.2857x

Now you can substitute this expression for y back into the first proportion to solve for x:

GB/CB = BE/x

GB/CB = ((GB/CB) * 0.2857x) / x

1 = 0.2857

x ≈ 321.42 ft

4. Find distance to P: Finally, you can use the expression for y to find the distance from P to E:

y ≈ 0.2857x ≈ 0.2857 * 321.42 ft ≈ 90.9 ft

Therefore, the distance from B to E is approximately 321.42 ft and the distance from P to E is approximately 90.9 ft.