Deltoid ABCD: Calculate DB When AC is 75% of DB and Area is 108X cm²

Q: What exactly is a deltoid and how is it different from other quadrilaterals?

A deltoid (also called a kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or squares, its diagonals are perpendicular and one bisects the other, which is why we can use the formula $ \frac{1}{2} \times d_1 \times d_2 $.

Q: Why does AC being 75% of DB matter for solving this problem?

This relationship lets us express both diagonals in terms of one variable! Instead of having AC and DB as separate unknowns, we can write AC = 0.75 × DB, reducing our problem to one equation with one unknown.

Q: How do I handle the 'X' in the area and the statement that DB = X?

The problem gives you two key facts: the area is 108X cm² and DB equals X cm. Substitute DB = X into your equation, then solve for the actual numerical value of X.

Q: Can I solve this problem using a different approach?

Yes! You could set up the equation as $ \frac{1}{2} \times \frac{3}{4}DB \times DB = 108X $ using fractions instead of decimals. Either way, you'll get $ \frac{3}{8}DB^2 = 108X $ and the same final answer.

Deltoid Area with Diagonal Relationships

Given the deltoid ABCD

Diagonal AC is equal to 75% of the diagonal DB

Area of the deltoid is equal to 108X cm².

Calculate the side DB,
if it was given that it is equal to X

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find side DB (X)

00:06 Side ratio according to the given data

00:17 Convert from percentage to fraction

00:22 Use the formula for calculating kite area

00:30 Product of diagonals divided by 2

00:34 Substitute appropriate values according to the data, and solve for X

00:50 Multiply by 2 to eliminate the fraction

00:59 Simplify what's possible

01:09 Isolate X

01:22 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the deltoid ABCD

Diagonal AC is equal to 75% of the diagonal DB

Area of the deltoid is equal to 108X cm².

Calculate the side DB,
if it was given that it is equal to X

Step-by-step solution

To solve this problem, let's proceed through the steps:

Step 1: Formula for the Area of a Deltoid

The area of a deltoid can be calculated through its diagonals using the formula:

$\text{Area} = \frac{1}{2} \times \text{Diagonal 1} \times \text{Diagonal 2}$

In this problem, the diagonals are $AC$ and $DB$ .

Step 2: Express $AC$ in terms of $DB$

We are told that $AC$ is $75\%$ of $DB$ . Therefore:

$AC = 0.75 \times DB$

Step 3: Substitute into the Area Formula

Substitute $AC = 0.75 \times DB$ into the area formula:

$\frac{1}{2} \times (0.75 \times DB) \times DB = 108X$

Simplifying gives:

$0.375 \times DB^2 = 108X$

Step 4: Solve for $DB$

Substituting $DB = X$ and rearranging the equation:

$0.375 \times X^2 = 108X$

Divide both sides by $X$ (assuming $X \neq 0$ ):

$0.375 X = 108$

Solving for $X$ :

$X = \frac{108}{0.375} = 288$

Thus, the length of diagonal $DB$ is $\boxed{288}$ cm.

Final Answer

288 cm²

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid = $\frac{1}{2} \times d_1 \times d_2$
Technique: Express AC as 0.75 × DB, then substitute into area formula
Check: Area = $\frac{1}{2} \times 216 \times 288 = 31,104$ when X = 288 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to use the diagonal relationship
Don't set up the area formula as $\frac{1}{2} \times AC \times DB = 108X$ without expressing AC in terms of DB = you'll have two unknowns! This makes the equation unsolvable. Always substitute AC = 0.75 × DB first to create one equation with one unknown.

Practice Quiz

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FAQ

Everything you need to know about this question

What exactly is a deltoid and how is it different from other quadrilaterals?

A deltoid (also called a kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or squares, its diagonals are perpendicular and one bisects the other, which is why we can use the formula $\frac{1}{2} \times d_1 \times d_2$ .

Why does AC being 75% of DB matter for solving this problem?

This relationship lets us express both diagonals in terms of one variable! Instead of having AC and DB as separate unknowns, we can write AC = 0.75 × DB, reducing our problem to one equation with one unknown.

How do I handle the 'X' in the area and the statement that DB = X?

The problem gives you two key facts: the area is 108X cm² and DB equals X cm. Substitute DB = X into your equation, then solve for the actual numerical value of X.

What if I get a different answer when I substitute back?

Double-check your algebra! Make sure you correctly converted 75% to 0.75, properly substituted DB = X, and divided $\frac{108}{0.375}$ correctly. The calculation should give exactly 288.

Can I solve this problem using a different approach?

Yes! You could set up the equation as $\frac{1}{2} \times \frac{3}{4}DB \times DB = 108X$ using fractions instead of decimals. Either way, you'll get $\frac{3}{8}DB^2 = 108X$ and the same final answer.

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Deltoid ABCD: Calculate DB When AC is 75% of DB and Area is 108X cm²

Deltoid Area with Diagonal Relationships

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What exactly is a deltoid and how is it different from other quadrilaterals?

Why does AC being 75% of DB matter for solving this problem?

How do I handle the 'X' in the area and the statement that DB = X?

What if I get a different answer when I substitute back?

Can I solve this problem using a different approach?

🌟 Unlock Your Math Potential

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