Question

directions: solve each problem to find the value of x to given that the quadrilateral is a rectangle. find its matching answer in the answer box and decode the answer to your work.
answer:
\frac{6}{7} \frac{12}{14} \frac{1}{10} \frac{7}{2} \frac{9}{7} \frac{4}{14} \frac{11}{8} \frac{7}{14} \frac{3}{4} \frac{7}{15} \frac{5}{9} \frac{2}{13}

1. x = ______

7x - 12
4x + 18

1. x = ______

(13x + 12)

1. x = ______

8x - 18
4x + 58

1. x = ______

14x - 9
16x - 35

1. x = ______

3x + 6
10x - 43

1. x = ______

(5x + 16)°
(3x - 62)°

Explanation:

Step1: Set equal rectangle sides

In a rectangle, opposite sides are congruent, so $7x - 12 = 4x + 18$.

Step2: Isolate x terms

Subtract $4x$ from both sides: $7x - 4x - 12 = 18$ → $3x - 12 = 18$.

Step3: Solve for x

Add 12 to both sides, then divide by 3: $3x = 18 + 12$ → $3x = 30$ → $x = \frac{30}{3} = 10$.

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Step1: Use rectangle right angle property

All angles in a rectangle are $90^\circ$, so $13x + 12 = 90$.

Step2: Isolate x term

Subtract 12 from both sides: $13x = 90 - 12$ → $13x = 78$.

Step3: Solve for x

Divide by 13: $x = \frac{78}{13} = 6$.

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Step1: Set equal rectangle diagonals

In a rectangle, diagonals are congruent, so $8x - 18 = 4x + 58$.

Step2: Isolate x terms

Subtract $4x$ from both sides: $8x - 4x - 18 = 58$ → $4x - 18 = 58$.

Step3: Solve for x

Add 18 to both sides, then divide by 4: $4x = 58 + 18$ → $4x = 76$ → $x = \frac{76}{4} = 19$.

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Step1: Set equal rectangle diagonals

In a rectangle, diagonals are congruent, so $14x - 9 = 16x - 35$.

Step2: Isolate x terms

Subtract $14x$ from both sides: $-9 = 2x - 35$.

Step3: Solve for x

Add 35 to both sides, then divide by 2: $2x = 35 - 9$ → $2x = 26$ → $x = \frac{26}{2} = 13$.

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Step1: Set equal rectangle sides

In a rectangle, opposite sides are congruent, so $3x + 6 = 10x - 43$.

Step2: Isolate x terms

Subtract $3x$ from both sides: $6 = 7x - 43$.

Step3: Solve for x

Add 43 to both sides, then divide by 7: $7x = 6 + 43$ → $7x = 49$ → $x = \frac{49}{7} = 7$.

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Step1: Use perpendicular diagonals property

The marked angles show perpendicular segments, so their sum is $90^\circ$: $(5x + 16) + (3x - 62) = 90$.

Step2: Simplify the equation

Combine like terms: $8x - 46 = 90$.

Step3: Solve for x

Add 46 to both sides, then divide by 8: $8x = 90 + 46$ → $8x = 136$ → $x = \frac{136}{8} = 17$.

Answer:

1. $x=10$
2. $x=6$
3. $x=19$
4. $x=13$
5. $x=7$
6. $x=17$