Question

the formula for the area of a rhombus is $a = \frac{1}{2}d_1d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals. which are equivalent equations? choose two correct answers. $d_1 = \frac{2a}{d_2}$ $d_2 = 2ad_1$ $d_2 = \frac{2a}{d_1}$ $d_2 = \frac{d_1}{2a}$

Explanation:

Step1: Start with the area formula of a rhombus

The given formula is \( A=\frac{1}{2}d_1d_2 \). We need to derive equivalent equations for \( d_1 \) and \( d_2 \) by isolating each variable.

Step2: Solve for \( d_1 \)

Multiply both sides of the equation \( A = \frac{1}{2}d_1d_2 \) by \( 2 \) to get rid of the fraction:
\( 2A=d_1d_2 \).
Then, divide both sides by \( d_2 \) (assuming \( d_2
eq0 \)) to solve for \( d_1 \):
\( d_1=\frac{2A}{d_2} \).

Step3: Solve for \( d_2 \)

Starting again from \( 2A = d_1d_2 \) (from Step 2), divide both sides by \( d_1 \) (assuming \( d_1
eq0 \)) to solve for \( d_2 \):
\( d_2=\frac{2A}{d_1} \).

Now, let's check the options:

• The first option \( d_1=\frac{2A}{d_2} \) matches our derivation for \( d_1 \).
• The third option \( d_2=\frac{2A}{d_1} \) matches our derivation for \( d_2 \).
• The second option \( d_2 = 2Ad_1 \) is incorrect (we derived \( d_2=\frac{2A}{d_1} \), not \( 2Ad_1 \)).
• The fourth option \( d_2=\frac{d_1}{2A} \) is incorrect (does not match our derivation).

Answer:

The two correct equivalent equations are:

• \( d_1=\frac{2A}{d_2} \)
• \( d_2=\frac{2A}{d_1} \)