Grades 9-10 Video Solutions 2023-2023_9-10

Video Transcription

The diagram shows three squares of side length 3cm, 5cm, and 8cm. What is the area in cm2 of the shaded trapezoid?  So let's call the vertical side of our trapezoid P and the right side Q, where P and Q are some length in cm. Let's use some similar triangles as well.  If we look at the biggest similar triangle, which would be this triangle right here,  we note that the right height is 8cm, and we can find that 3 plus 5 plus 8 is 16, which is the bottom length.  So the ratio of the short side to the long side is 1 to 2 in these similar triangles.  Therefore, the ratio of this side, which we called Q, over this side, which we can find to be 8, also needs to be 1 to 2, and we can find that Q equals 4. We do the same right here.  We know this is P, and P over 3 has to equal half. Then we know P is 3 halves.  Now we can use our area of a trapezoid formula, knowing base 1 plus base 2 divided by 2 times height is area.  So we find that the first base is a side length of 4, and the second is 3 halves, and we also know that the height of this trapezoid is 5,  and then dividing by 2, we know that 4 plus 3 halves is 11 halves, and 11 halves times 5 halves is 55 over 4, giving the correct answer, B.

Video Summary

The video explains how to calculate the area of a shaded trapezoid formed from three squares with side lengths of 3cm, 5cm, and 8cm. By leveraging similar triangles, it determines that the lengths of the vertical side (P) and right side (Q) are \( \frac{3}{2} \) and 4 cm, respectively. Using the trapezoid area formula, where the area is calculated as \(( \text{base}_1 + \text{base}_2 ) / 2 \times \text{height}\), they determine that with a height of 5 cm, the area is \( \frac{55}{4} \) cm\(^2\). The solution confirms the correct area answer is choice B.

Keywords

trapezoid area

similar triangles

squares side lengths

geometry calculation

shaded trapezoid