Question

if ge = 79, ef = 70, gf = 84, hi = 35, and ji = 42, find the perimeter of △hij. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Check Triangle Similarity

First, check the angles of both triangles. For $\triangle GEF$: angles are $65^\circ$, $68^\circ$, and $180 - 65 - 68 = 47^\circ$? Wait, no, looking at the diagram, $\angle G = 65^\circ$, $\angle E = 68^\circ$, so $\angle F = 180 - 65 - 68 = 47^\circ$? Wait, no, the other triangle $\triangle HIJ$ has $\angle J = 51^\circ$? Wait, no, maybe I misread. Wait, the first triangle: $\angle G = 65^\circ$, $\angle E = 68^\circ$, so $\angle F = 180 - 65 - 68 = 47^\circ$? No, wait the second triangle: $\angle I = 61^\circ$, $\angle H = 68^\circ$, so $\angle J = 180 - 61 - 68 = 51^\circ$. Wait, maybe the first triangle: $\angle G = 65^\circ$, $\angle F = 61^\circ$? Wait, the diagram: first triangle (left) has $\angle E = 68^\circ$, $\angle G = 65^\circ$, so $\angle F = 180 - 68 - 65 = 47^\circ$? No, that doesn't match. Wait, maybe the triangles are similar by AA similarity. Let's check angles:

In $\triangle GEF$: $\angle E = 68^\circ$, $\angle G = 65^\circ$, so $\angle F = 180 - 68 - 65 = 47^\circ$? No, the right triangle $\triangle HIJ$: $\angle H = 68^\circ$, $\angle I = 61^\circ$, so $\angle J = 180 - 68 - 61 = 51^\circ$. Wait, maybe I made a mistake. Wait, the first triangle: $\angle E = 68^\circ$, $\angle F = 61^\circ$, so $\angle G = 180 - 68 - 61 = 51^\circ$? Ah, that makes sense. So $\angle G = 51^\circ$, $\angle E = 68^\circ$, $\angle F = 61^\circ$. Then $\triangle HIJ$ has $\angle J = 51^\circ$, $\angle H = 68^\circ$, $\angle I = 61^\circ$. So by AA similarity (two angles equal), $\triangle GEF \sim \triangle HIJ$ (since $\angle E = \angle H = 68^\circ$, $\angle F = \angle I = 61^\circ$, so the third angles $\angle G = \angle J = 51^\circ$). So similarity ratio:

First, find the sides of $\triangle GEF$: $GE = 79$, $EF = 70$, $GF = 84$. Wait, no, $EF = 70$, $GF = 84$, $GE = 79$? Wait, no, maybe the sides: in $\triangle GEF$, $EF = 70$, $GF = 84$, $GE = 79$? Wait, no, the corresponding sides: in similar triangles, the ratio of sides is equal. Let's see, in $\triangle HIJ$, $HI = 35$, $JI = 42$. Let's find the ratio of similarity.

In $\triangle GEF$, $EF = 70$, and in $\triangle HIJ$, $HI = 35$. So the ratio of $\triangle HIJ$ to $\triangle GEF$ is $35/70 = 1/2$. Let's check another side: $JI = 42$, and in $\triangle GEF$, $GF = 84$. So $42/84 = 1/2$. So the similarity ratio is $1/2$ (since $\triangle HIJ$ is similar to $\triangle GEF$ with ratio $1/2$).

Now, we need to find the third side of $\triangle HIJ$, which corresponds to $GE$ in $\triangle GEF$. $GE = 79$, so the corresponding side $HJ$ (since in $\triangle GEF$, $GE$ is opposite $\angle F$, and in $\triangle HIJ$, $HJ$ is opposite $\angle I$; wait, maybe better to list corresponding sides:

Since $\angle E = \angle H = 68^\circ$, $\angle F = \angle I = 61^\circ$, $\angle G = \angle J = 51^\circ$, so:

- $\angle E$ (68°) corresponds to $\angle H$ (68°), so side $EF$ (opposite $\angle G$) corresponds to side $HI$ (opposite $\angle J$)
- $\angle F$ (61°) corresponds to $\angle I$ (61°), so side $GF$ (opposite $\angle E$) corresponds to side $JI$ (opposite $\angle H$)
- $\angle G$ (51°) corresponds to $\angle J$ (51°), so side $GE$ (opposite $\angle F$) corresponds to side $HJ$ (opposite $\angle I$)

So $EF$ (70) corresponds to $HI$ (35) → ratio 35/70 = 1/2

$GF$ (84) corresponds to $JI$ (42) → ratio 42/84 = 1/2

So $GE$ (79) corresponds to $HJ$ → $HJ = GE \times (1/2) = 79 \times 0.5 = 39.5$

Now, perimeter of $\triangle HIJ$ is $HI + JI + HJ = 35 + 42 + 39.5 = 116.5$? Wait, no, wait:

Wait, $EF = 70$ (side of $\triangle GEF$), $HI = 35$ (side of $\triangle HIJ$) → ratio is $HI/EF = 35/70 = 1/2$

$GF = 84$ (side of $\triangle GEF$), $JI = 42$ (side of $\triangle HIJ$) → ratio $JI/GF = 42/84 = 1/2$

So the ratio of $\triangle HIJ$ to $\triangle GEF$ is $1/2$. Therefore, the third side of $\triangle HIJ$ (corresponding to $GE$) is $GE \times (1/2) = 79 \times 0.5 = 39.5$

Now, perimeter of $\triangle HIJ$: $35 + 42 + 39.5 = 116.5$? Wait, but let's check again. Wait, maybe the sides:

In $\triangle GEF$: sides are $EF = 70$, $GF = 84$, $GE = 79$

In $\triangle HIJ$: sides are $HI = 35$ (corresponds to $EF$), $JI = 42$ (corresponds to $GF$), and $HJ$ (corresponds to $GE$)

So ratio is $35/70 = 1/2$, so $HJ = GE \times (1/2) = 79 \times 0.5 = 39.5$

Then perimeter is $35 + 42 + 39.5 = 116.5$

Wait, but let's confirm the similarity. Angles:

$\triangle GEF$: $\angle E = 68^\circ$, $\angle F = 61^\circ$, $\angle G = 51^\circ$

$\triangle HIJ$: $\angle H = 68^\circ$, $\angle I = 61^\circ$, $\angle J = 51^\circ$

So AA similarity (two angles equal: $\angle E = \angle H = 68^\circ$, $\angle F = \angle I = 61^\circ$), so triangles are similar with ratio $1/2$ (since $HI = 35 = 70 \times 1/2$, $JI = 42 = 84 \times 1/2$)

Thus, the third side $HJ = GE \times 1/2 = 79 \times 0.5 = 39.5$

Perimeter: $35 + 42 + 39.5 = 116.5$

Step2: Calculate Perimeter

Sum the three sides of $\triangle HIJ$: $HI = 35$, $JI = 42$, $HJ = 39.5$

Perimeter = $35 + 42 + 39.5 = 116.5$

Answer:

116.5