Question

the figure shown is a rectangle. solve for x. *

Explanation:

Step1: Recall rectangle diagonal property

In a rectangle, the diagonals are equal and bisect each other, so the triangles formed are isosceles. Also, alternate interior angles (or in this case, the angles formed by the diagonal and the sides) are equal when considering the properties of rectangles (opposite sides are parallel, so alternate interior angles are equal). So we set the two angle expressions equal: $4x - 12 = x + 2$.

Step2: Solve for x

Subtract $x$ from both sides: $4x - x - 12 = x - x + 2$ which simplifies to $3x - 12 = 2$. Then add 12 to both sides: $3x - 12 + 12 = 2 + 12$ which gives $3x = 14$? Wait, no, wait. Wait, maybe I made a mistake. Wait, in a rectangle, the diagonal splits the rectangle into two congruent triangles, and also, the angles formed by the diagonal and a side: actually, in a rectangle, the diagonals are equal and bisect each other, so the triangles are isosceles, but also, the base angles of the isosceles triangle? Wait, no, maybe the angles are equal because the diagonal is a transversal between the two parallel sides, so alternate interior angles are equal. So the angle $(4x - 12)^\circ$ and $(x + 2)^\circ$ are equal. So:

$4x - 12 = x + 2$

Subtract $x$ from both sides: $3x - 12 = 2$

Add 12 to both sides: $3x = 14$? No, that can't be. Wait, maybe I misread the angles. Wait, maybe the angles are complementary? Wait, no, in a rectangle, the corner is a right angle (90 degrees). Wait, maybe the two angles are part of the right angle. Oh! Wait, that's probably it. The diagonal splits the right angle into two angles, so their sum is 90 degrees. Oh! I made a mistake earlier. So $(4x - 12) + (x + 2) = 90$. That makes more sense. Let's correct that.

Step1 (corrected): Sum of angles is 90

In a rectangle, each interior angle is $90^\circ$. The diagonal divides the right angle into two angles, so their sum is $90^\circ$. So:

$(4x - 12) + (x + 2) = 90$

Step2 (corrected): Simplify the equation

Combine like terms: $4x + x - 12 + 2 = 90$ → $5x - 10 = 90$

Step3 (corrected): Solve for x

Add 10 to both sides: $5x - 10 + 10 = 90 + 10$ → $5x = 100$

Divide both sides by 5: $x = \frac{100}{5} = 20$

Wait, that makes sense. So my initial mistake was not realizing that the two angles are complementary (sum to 90 degrees) because they are part of the right angle in the rectangle. Let's verify:

If $x = 20$, then $4x - 12 = 4*20 - 12 = 80 - 12 = 68$ and $x + 2 = 20 + 2 = 22$. Then $68 + 22 = 90$, which is the right angle. Perfect, that works. So the correct equation is the sum of the two angles equals 90.

So let's redo the steps correctly:

Step1: Set up the equation

Since the angle is a right angle ($90^\circ$), the sum of the two angles is $90^\circ$:

$4x - 12 + x + 2 = 90$

Step2: Combine like terms

$5x - 10 = 90$

Step3: Solve for x

Add 10 to both sides: $5x = 100$

Divide by 5: $x = 20$

Answer:

$x = 20$