Question

1. the two triangles below are similar.

image of two right triangles, one with legs 5 (vertical) and hypotenuse 14 - x, the other with leg 3 (vertical) and hypotenuse x + 7, sharing a common vertex h

find the value of x.
enter each line of work as an equation.
input box: enter your next step here
math symbols: +-*/, ≥, (), π, √, a^b, a/b

Explanation:

Step1: Set up proportion (similar triangles)

Since the triangles are similar, the ratios of corresponding sides are equal. So, $\frac{x + 7}{14 - x}=\frac{3}{5}$.

Step2: Cross - multiply

Cross - multiplying gives $5(x + 7)=3(14 - x)$.

Step3: Expand both sides

Expanding, we get $5x+35 = 42-3x$.

Step4: Add 3x to both sides

Adding $3x$ to both sides: $5x + 3x+35=42-3x + 3x$, which simplifies to $8x+35 = 42$.

Step5: Subtract 35 from both sides

Subtracting 35 from both sides: $8x+35 - 35=42 - 35$, so $8x=7$.

Step6: Divide by 8

Dividing both sides by 8: $x=\frac{7}{8}=0.875$. Wait, there is a mistake in the proportion setup. Let's re - examine the similar triangles.

Actually, the corresponding sides should be $\frac{x + 7}{14 - x}=\frac{3}{5}$ is wrong. Let's identify the corresponding sides correctly. Triangle $FGH$ and triangle $IJH$ are similar (right - angled triangles, and $\angle FHG=\angle JHI$ (vertical angles), so by AA similarity). So the sides: $FG = 3$, $IG = 5$, $FH=x + 7$, $JH=14 - x$. Wait, no, the correct proportion is $\frac{FH}{JH}=\frac{FG}{IJ}$. So $FH=x + 7$, $JH = 14 - x$, $FG = 3$, $IJ = 5$. So the correct proportion is $\frac{x + 7}{14 - x}=\frac{3}{5}$ was wrong. Wait, $FG$ is 3, $IJ$ is 5, $FH$ is $x + 7$, $JH$ is $14 - x$. Wait, actually, the two right triangles: triangle $FGH$ with legs 3 and $FG$ (wait, no, $FG$ is a leg, length 3, and $FH$ is the hypotenuse $x + 7$; triangle $IJH$ with leg 5 and hypotenuse $14 - x$. Wait, no, the legs are $FG = 3$, $GH$ (horizontal leg, not given), $IJ = 5$, $IH$ (vertical leg, length 5), and $JH$ (hypotenuse $14 - x$), $FH$ (hypotenuse $x + 7$). Wait, no, the correct proportion is $\frac{FG}{IJ}=\frac{FH}{JH}$, so $\frac{3}{5}=\frac{x + 7}{14 - x}$ is incorrect. Wait, let's use the segments of the hypotenuse and the legs. Wait, actually, the two triangles are similar, so the ratio of the legs is equal to the ratio of the hypotenuses. So $\frac{3}{5}=\frac{x + 7}{14 - x}$ is wrong. Let's do it again.

Let's consider the two similar triangles: $\triangle FGH\sim\triangle IJH$. So $\angle F=\angle J$ (since $\triangle FGH$ and $\triangle IJH$ are similar), so the sides: $FG$ corresponds to $IJ$, $GH$ corresponds to $IH$, and $FH$ corresponds to $JH$. So $FG = 3$, $IJ = 5$, $FH=x + 7$, $JH = 14 - x$. So the correct proportion is $\frac{FG}{IJ}=\frac{FH}{JH}$, so $\frac{3}{5}=\frac{x + 7}{14 - x}$ is incorrect. Wait, no, $FG$ is a leg, $IJ$ is a leg, $FH$ is a hypotenuse, $JH$ is a hypotenuse. So the ratio of legs should equal the ratio of hypotenuses. So $\frac{3}{5}=\frac{x + 7}{14 - x}$ is wrong. Wait, let's use the correct corresponding sides. Let's look at the vertical segments: $IG = 5$, $FG = 3$, and the hypotenuses: $FH=x + 7$, $JH = 14 - x$. Wait, the two triangles share the angle at $H$, so by AA similarity (right angle and vertical angle), so $\triangle FGH\sim\triangle IJH$. Therefore, $\frac{FG}{IJ}=\frac{FH}{JH}$, so $\frac{3}{5}=\frac{x + 7}{14 - x}$ is incorrect. Wait, $FG$ is 3 (leg of $\triangle FGH$), $IJ$ is 5 (leg of $\triangle IJH$), $FH$ is hypotenuse of $\triangle FGH$ ($x + 7$), $JH$ is hypotenuse of $\triangle IJH$ ($14 - x$). So the ratio of legs is $\frac{3}{5}$, and the ratio of hypotenuses is $\frac{x + 7}{14 - x}$. So they should be equal: $\frac{3}{5}=\frac{x + 7}{14 - x}$. But when we solve this:

Cross - multiply: $3(14 - x)=5(x + 7)$

$42-3x = 5x+35$

$42-35=5x + 3x$

$7 = 8x$

$x=\frac{7}{8}=0.875$. But let's check with the correct proportion. Wait, maybe the sides are $FG = 3$, $IJ = 5$, $FH=x + 7$, $JH = 14 - x$, and the other legs (horizontal) are equal? No, the horizontal legs are not given. Wait, maybe the proportion is $\frac{x + 7}{14 - x}=\frac{3}{5}$ is correct. Let's re - solve:

$3(14 - x)=5(x + 7)$

$42-3x=5x + 35$

$42-35 = 5x+3x$

$7 = 8x$

$x=\frac{7}{8}=0.875$. But let's check the lengths. If $x=\frac{7}{8}$, then $FH=\frac{7}{8}+7=\frac{7 + 56}{8}=\frac{63}{8}=7.875$, $JH=14-\frac{7}{8}=\frac{112 - 7}{8}=\frac{105}{8}=13.125$. Then $\frac{FH}{JH}=\frac{63/8}{105/8}=\frac{63}{105}=\frac{3}{5}$, which matches $\frac{FG}{IJ}=\frac{3}{5}$. So the solution is correct.

Answer:

$x = \frac{7}{8}$ (or $x = 0.875$)