Question

1. δmkl is an isosceles triangle with \\(\overline{mk} \cong \overline{ml}\\). if \\(mk = 7x - 15\\), \\(kl = 4x - 5\\), and \\(ml = 10x - 42\\), find x and the measure of each side.\
2. \\(\triangle abc\\) is an isosceles triangle with \\(\overline{ab} \cong \overline{bc}\\). if \\(ab = 3x + 41\\), \\(bc = 8x - 54\\), and \\(ac = 12x - 103\\), find x and the measure of each side.\
3. \\(\triangle ghi\\) is an isosceles triangle with \\(\overline{gi} \cong \overline{hi}\\). if \\(gh\\) is three more than x, \\(hi\\) is 17 less than four times x, and \\(gi\\) is 45 less than six times x, find x and the measure of each side.

Explanation:

Problem 9: $\triangle ABC$ is isosceles with $AB \cong BC$.

Step 1: Set $AB = BC$

Since $AB \cong BC$, their lengths are equal. So, $3x + 41 = 8x - 54$.

Step 2: Solve for $x$

Subtract $3x$ from both sides: $41 = 5x - 54$.
Add $54$ to both sides: $95 = 5x$.
Divide by $5$: $x = 19$.

Step 3: Find side lengths

• $AB = 3(19) + 41 = 57 + 41 = 98$.
• $BC = 8(19) - 54 = 152 - 54 = 98$.
• $AC = 12(19) - 103 = 228 - 103 = 125$.

Problem 8: $\triangle MKL$ is isosceles with $MK \cong ML$.

Step 1: Set $MK = ML$

Since $MK \cong ML$, $7x - 15 = 10x - 42$.

Step 2: Solve for $x$

Subtract $7x$: $-15 = 3x - 42$.
Add $42$: $27 = 3x$.
Divide by $3$: $x = 9$.

Step 3: Find side lengths

• $MK = 7(9) - 15 = 63 - 15 = 48$.
• $ML = 10(9) - 42 = 90 - 42 = 48$.
• $KL = 4(9) - 5 = 36 - 5 = 31$.

Problem 10: $\triangle GHI$ is isosceles with $GI \cong HI$.

Step 1: Define side expressions

• $GH = x + 3$ (3 more than $x$).
• $HI = 4x - 17$ (17 less than $4x$).
• $GI = 6x - 45$ (45 less than $6x$).

Since $GI \cong HI$, $6x - 45 = 4x - 17$.

Step 2: Solve for $x$

Subtract $4x$: $2x - 45 = -17$.
Add $45$: $2x = 28$.
Divide by $2$: $x = 14$.

Step 3: Find side lengths

• $GH = 14 + 3 = 17$.
• $HI = 4(14) - 17 = 56 - 17 = 39$.
• $GI = 6(14) - 45 = 84 - 45 = 39$.

Final Answers:

Problem 9:

$x = 19$, $AB = 98$, $BC = 98$, $AC = 125$.

Problem 8:

$x = 9$, $MK = 48$, $ML = 48$, $KL = 31$.

Problem 10:

$x = 14$, $GH = 17$, $HI = 39$, $GI = 39$.

Answer:

Step 1: Define side expressions

• $GH = x + 3$ (3 more than $x$).
• $HI = 4x - 17$ (17 less than $4x$).
• $GI = 6x - 45$ (45 less than $6x$).

Since $GI \cong HI$, $6x - 45 = 4x - 17$.

Step 2: Solve for $x$

Subtract $4x$: $2x - 45 = -17$.
Add $45$: $2x = 28$.
Divide by $2$: $x = 14$.

Step 3: Find side lengths

• $GH = 14 + 3 = 17$.
• $HI = 4(14) - 17 = 56 - 17 = 39$.
• $GI = 6(14) - 45 = 84 - 45 = 39$.

Final Answers:

Problem 9:

$x = 19$, $AB = 98$, $BC = 98$, $AC = 125$.

Problem 8:

$x = 9$, $MK = 48$, $ML = 48$, $KL = 31$.

Problem 10:

$x = 14$, $GH = 17$, $HI = 39$, $GI = 39$.