Question

given: parallelogram abcd is a rhombus. side bc = 10 and segment ao = 6. answer the questions below. (2 points each)1. true diagonal bd bisects angle d.2. what is the measure of angle $d_1$?$74^circ$3. false diagonal bd bisects diagonal ac.4. what is the measure of segment oc?6 cm5. false diagonal bd and ac are congruent.6. true diagonals bd and ac are perpendicular.7. true triangle boc is a right triangle.8. what is the measure of diagonal bd? show your work. click here to watch a help video.16 cm9. what is the area of triangle boc? show your work. click here to watch a help video.10. what is the area of rhombus abcd? show your work. click here to watch a help video.

Explanation:

Step1: Find length of BO

In right $\triangle BOC$, use Pythagorean theorem:
$$BO = \sqrt{BC^2 - OC^2}$$
We know $BC=10$ cm, $OC=AO=6$ cm, so:
$$BO = \sqrt{10^2 - 6^2} = \sqrt{100-36} = \sqrt{64} = 8 \text{ cm}$$

Step2: Calculate diagonal BD

Diagonals of rhombus bisect each other, so $BD=2\times BO$:
$$BD = 2\times8 = 16 \text{ cm}$$

Step3: Calculate area of $\triangle BOC$

Area of right triangle: $\frac{1}{2}\times base\times height$
$$Area_{\triangle BOC} = \frac{1}{2}\times BO\times OC = \frac{1}{2}\times8\times6 = 24 \text{ cm}^2$$

Step4: Calculate area of rhombus ABCD

Area of rhombus is 4 times area of $\triangle BOC$ (or $\frac{1}{2}\times d_1\times d_2$):
$$Area_{ABCD} = 4\times24 = 96 \text{ cm}^2$$
Or using diagonals: $AC=12$ cm, $BD=16$ cm:
$$Area_{ABCD} = \frac{1}{2}\times12\times16 = 96 \text{ cm}^2$$

Answer:

1. True
2. $74^\circ$
3. False
4. 6 cm
5. False
6. True
7. True
8. 16 cm
9. 24 $\text{cm}^2$
10. 96 $\text{cm}^2$