Right Triangle Area Problem: Solve for X When Area = 6 cm²

Q: How do I know which sides are the legs in a right triangle?

The legs are the two sides that meet at the right angle (90°). The hypotenuse is always the longest side, opposite the right angle. In this problem, sides of length 4 and (X-1) are the legs.

Q: Why do we multiply both sides by 2 when solving the area equation?

When we have $ \frac{4(X-1)}{2} = 6 $, multiplying both sides by 2 eliminates the fraction. This gives us $ 4(X-1) = 12 $, which is much easier to solve!

Q: How can I double-check that BC = X-1 = 3?

Substitute X=4: BC = 4-1 = 3. Then verify the area: $ \frac{4 \times 3}{2} = \frac{12}{2} = 6 $ cm² ✓

Q: Why isn't the hypotenuse (X+1) used in the area calculation?

The area formula for right triangles uses only the perpendicular sides (legs). The hypotenuse is at an angle, so it doesn't contribute directly to the rectangular area calculation.

Triangle Area with Variable Expressions

Triangle ABC is a right triangle.

The area of the triangle is 6 cm².

Calculate X and the length of the side BC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the value of X and calculate BC

00:06 (height(AC) X base (BC)) divided by 2

00:14 Substitute in the relevant values

00:24 Calculate and solve

00:32 Open the parentheses from left to right

00:40 Isolate X

00:47 This is the size of X

00:49 Q.E.D.1

00:52 Substitute X in side BC

00:59 This is the size of BC

01:01 Q.E.D.2

01:04 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Triangle ABC is a right triangle.

The area of the triangle is 6 cm².

Calculate X and the length of the side BC.

Step-by-step solution

We use the formula to calculate the area of the right triangle:

$\frac{AC\cdot BC}{2}=\frac{cateto\times cateto}{2}$

And compare the expression with the area of the triangle $6$

$\frac{4\cdot(X-1)}{2}=6$

Multiplying the equation by the common denominator means that we multiply by $2$

$4(X-1)=12$

We distribute the parentheses before the distributive property

$4X-4=12$ / $+4$

$4X=16$ / $:4$

$X=4$

We replace $X=4$ in the expression $BC$ and

find:

$BC=X-1=4-1=3$

Final Answer

X=4, BC=3

Key Points to Remember

Essential concepts to master this topic

Area Formula: For right triangles, Area = $\frac{1}{2} \times \text{base} \times \text{height}$
Setup Equation: $\frac{4 \times (X-1)}{2} = 6$ leads to $4(X-1) = 12$
Verify Answer: Check X=4 gives BC=3, then Area = $\frac{4 \times 3}{2} = 6$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong sides in area formula
Don't use the hypotenuse (X+1) in the area calculation = wrong equation! The hypotenuse is never used for right triangle area. Always use only the two perpendicular sides (legs) in the area formula.

Practice Quiz

Test your knowledge with interactive questions

Is the triangle in the drawing a right triangle?

FAQ

Everything you need to know about this question

How do I know which sides are the legs in a right triangle?

The legs are the two sides that meet at the right angle (90°). The hypotenuse is always the longest side, opposite the right angle. In this problem, sides of length 4 and (X-1) are the legs.

Why do we multiply both sides by 2 when solving the area equation?

When we have $\frac{4(X-1)}{2} = 6$ , multiplying both sides by 2 eliminates the fraction. This gives us $4(X-1) = 12$ , which is much easier to solve!

What if I get a negative value for X?

Check your algebra! Also remember that in geometry problems, lengths must be positive. If X gives a negative length for any side, that solution doesn't make sense geometrically.

How can I double-check that BC = X-1 = 3?

Substitute X=4: BC = 4-1 = 3. Then verify the area: $\frac{4 \times 3}{2} = \frac{12}{2} = 6$ cm² ✓

Why isn't the hypotenuse (X+1) used in the area calculation?

The area formula for right triangles uses only the perpendicular sides (legs). The hypotenuse is at an angle, so it doesn't contribute directly to the rectangular area calculation.

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Right Triangle Area Problem: Solve for X When Area = 6 cm²

Triangle Area with Variable Expressions

❤️ 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

How do I know which sides are the legs in a right triangle?

Why do we multiply both sides by 2 when solving the area equation?

What if I get a negative value for X?

How can I double-check that BC = X-1 = 3?

Why isn't the hypotenuse (X+1) used in the area calculation?

🌟 Unlock Your Math Potential

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