Imagine two triangles with the top triangle touching the bottom one. Both are sharing the point "S" as well. The bottom triangle has the point R on its left corner and point Q on its right corner. On the left side of the bottom triangle is "49" and on the right "84"

Question

For the top triangle, its left corner has Point G and its right corner Point H. On the left is "14" and on the right is "24"

Are the triangles similar, if so why? What would be the correct similarity statement if similar?
The triangles ___ (options: Are, Aren't) similar. The reason they are similar is ___ (Options: Aren't similar, SAS, AA, SSS). the similarity statement would be ___ (options: triangle GHS ~ Triangle RQS, Aren't similar, Triangle GHS ~ Triangle QRS, Triangle GHS ~ Triangle QSR

Answer 1

To determine if the two triangles are similar, we can use the information provided about their sides.

The bottom triangle has the lengths of the sides as follows:

• Left side (RQ) = 49
• Right side (QS) = 84

The top triangle has the lengths of the sides as follows:

• Left side (GH) = 14
• Right side (HS) = 24

We can check whether the triangles are similar by comparing corresponding sides.

For similarity, we need to see if the ratios of the lengths of corresponding sides are equal.

1. Comparing the left sides: \[ \frac{GH}{RQ} = \frac{14}{49} = \frac{2}{7} \]

2. Comparing the right sides: \[ \frac{HS}{QS} = \frac{24}{84} = \frac{2}{7} \]

Since both ratios are equal (\(\frac{2}{7}\)), the triangles are indeed similar by the Side-Angle-Side (SAS) similarity criterion.

Therefore, the triangles are similar, and the correct statements would be:

The triangles are similar. The reason they are similar is SAS. The similarity statement would be Triangle GHS ~ Triangle RQS.