Question

not drawn to scale (8x - 5) in. (3x + 10) in. the perimeter of parallelogram abcd is 46 inches. what is da? 3 in. 4 in. 8 in. 19 in.

Explanation:

Step1: Recall properties of parallelogram

In a parallelogram, opposite sides are equal. So \( AB = CD \) and \( AD = BC \). Given \( AB=(8x - 5) \) in and \( CD=(3x + 10) \) in. So we set \( 8x-5=3x + 10 \).

Step2: Solve for x

Subtract \( 3x \) from both sides: \( 8x-3x-5=3x-3x + 10 \) which simplifies to \( 5x-5 = 10 \). Then add 5 to both sides: \( 5x-5 + 5=10 + 5 \), so \( 5x=15 \). Divide both sides by 5: \( x=\frac{15}{5}=3 \).

Step3: Find length of AB (and CD)

Substitute \( x = 3 \) into \( AB = 8x-5 \): \( 8(3)-5=24 - 5=19 \) in. So \( AB = CD = 19 \) in.

Step4: Use perimeter formula

Perimeter of parallelogram \( P=2(AB + AD) \). Given \( P = 46 \) in. So \( 46=2(19+AD) \). Divide both sides by 2: \( 23=19 + AD \). Subtract 19 from both sides: \( AD=23 - 19 = 4 \) in? Wait, no, wait. Wait, maybe I mixed up. Wait, wait, the perimeter is \( 2(AB + BC) \), but in the diagram, \( AD \) and \( BC \) are the other sides. Wait, no, let's re - check. Wait, the sides are \( AB \) and \( AD \) as adjacent sides. Wait, the problem says "What is DA?". Wait, maybe I made a mistake. Wait, let's re - do.

Wait, in parallelogram \( ABCD \), \( AB \) and \( CD \) are opposite, \( AD \) and \( BC \) are opposite. Wait, the perimeter is \( 2(AB + AD) \). We found \( AB = 19 \) in? Wait, no, that can't be, because if \( x = 3 \), \( 8x-5=19 \), \( 3x + 10=19 \), so \( AB = CD = 19 \) in. Then perimeter is \( 2(AB + AD)=46 \), so \( AB + AD=23 \), so \( AD=23 - AB=23 - 19 = 4 \) in? But the options have 4 in as an option. Wait, but let's check again.

Wait, maybe the sides are \( AB=(8x - 5) \) and \( AD \) is the other side. Wait, the diagram: \( AB \) is \( (8x - 5) \), \( DC \) is \( (3x + 10) \), so \( AB = DC \), so \( 8x-5=3x + 10 \), \( 5x=15 \), \( x = 3 \). Then \( AB=8*3 - 5=19 \), \( DC = 3*3+10 = 19 \). Perimeter is \( 2(AB + AD)=46 \), so \( AB + AD=23 \), so \( AD=23 - 19 = 4 \) in. So \( DA = 4 \) in.

Wait, but the options are 3 in, 4 in, 8 in, 19 in. So the answer is 4 in.

Answer:

4 in.