Area of a Triangle: Using variables

To find $x$, the vertical height of the triangle, we will use the area formula for a triangle:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

We know that:

• The area of the triangle is 15.
• The base of the triangle, $BE$, is 5 units.
• The height of the triangle, $AE$, is $x$ units.

Substituting these values into the formula, we get:

$15 = \frac{1}{2} \times 5 \times x$

First, simplify the right side of the equation:

$15 = \frac{5}{2} \times x$

To isolate $x$, multiply both sides by 2:

$30 = 5x$

Finally, divide both sides by 5 to solve for $x$:

$x = \frac{30}{5} = 6$

Therefore, the value of $x$ is 6.

To solve for $x$, let's apply the standard formula for the area of a triangle:

• Given that the area $A = 15$, base $b = 3$, and height $h = x$.

The area formula is:

$A = \frac{1}{2} \times b \times h$

Substituting the given values into the equation, we have:

$15 = \frac{1}{2} \times 3 \times x$

Now, simplify and solve for $x$:

$15 = \frac{3}{2} \times x$

Multiply both sides by $\frac{2}{3}$ to isolate $x$:

$x = 15 \times \frac{2}{3}$

Calculating, we obtain:

$x = \frac{30}{3} = 10$

Thus, the height $x$ of the triangle is $x = 10$.

Therefore, the solution to the problem is $x = 10$.

To solve for $x$, we begin by applying the formula for the area of a triangle:

Given: the area is 18, AE is the height (6) , and EC is the x.

Insert the known values into the formula:

Simplify the equation:

Next, solve for $x$ by dividing both sides by 3:

Calculate:

Thus, the length $x$ is $\mathbf{6}$.

To solve this problem, let's apply the following steps:

• Step 1: Identify the formula for the area of a triangle, which is $A = \frac{1}{2} \times \text{base} \times \text{height}$.
• Step 2: Substitute the known values into the formula: $21 = \frac{1}{2} \times 7 \times x$.
• Step 3: Simplify and solve the equation for $x$.

Now, let's work through each step more precisely:
Step 1: We're given the area formula as $A = \frac{1}{2} \times b \times h$.
Step 2: Substitute in the known values: the area $A = 21$, the base $b = 7$, and the height $h = x$, leading to the equation $21 = \frac{1}{2} \times 7 \times x$.
Step 3: Solve for $x$ – first simplify the multiplication on the right: $21 = \frac{7}{2} \times x$.
Step 4: To isolate $x$, multiply both sides by 2 to get $42 = 7x$.
Step 5: Finally, divide both sides by 7 to solve for $x$: $x = \frac{42}{7} = 6$.

Therefore, the value of $x$ is $6$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given base and area of the triangle.
• Step 2: Apply the triangle area formula to find the height $x$.
• Step 3: Perform the necessary calculations to determine $x$.

Now, let's work through each step:
Step 1: The problem gives us the base $BC = 3$ and the area of the triangle as $9$.
Step 2: We'll use the formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
Step 3: Plugging in our values, the equation becomes $9 = \frac{1}{2} \times 3 \times x$.
Rearranging for $x$, we have: $x = \frac{2 \times 9}{3} = \frac{18}{3} = 6$.

Thus, the solution to the problem is $x = 6$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information.
• Step 2: Apply the formula for the area of a triangle.
• Step 3: Solve for $X$.

Now, let's work through each step:
Step 1: The problem provides the area of the triangle as $3$ square units, with the base of the triangle as $2$ units.
Step 2: The formula used for the area of a triangle is $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
Given that the base is $2$ and the area is $3$, we rearrange to find $X$ (the height):

$3 = \frac{1}{2} \times 2 \times X$

Simplifying, we get:

$3 = X$

Therefore, the length of $X$ is $3$.

To solve this problem, we'll use the formula for the area of a triangle:

• Step 1: Identify the given information: Area = 12, base $BC = 3$, height $AE = x$.
• Step 2: Use the area formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
• Step 3: Solve for $x$ (height) using $x = \frac{2 \times \text{Area}}{\text{base}}$.

Now, substituting the known values into the equation, we get:

$x = \frac{2 \times 12}{3}$

Performing the multiplication and division yields:

$x = \frac{24}{3} = 8$

Therefore, the length of $x$ is 8.

To solve this problem, we need to find the value of $x$, given that the area of the triangle is 16 and the base is known to be 4.

• Step 1: Identify known values.
  We know the area $A = 16$ and base $b = 4$.
• Step 2: Apply the triangle area formula:
  $A = \frac{1}{2} \times b \times h$, where $h$ is the height we need to calculate.
• Step 3: Solve for height $x$.
  Substitute values into the formula: $16 = \frac{1}{2} \times 4 \times x$.
• Step 4: Perform the necessary calculations:
  Simplify the equation: $16 = 2 \times x$.
  Divide both sides by 2 to solve for $x$.

The calculation simplifies to $x = 8$.

Therefore, the solution to the problem is $x = 8$.

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:

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

To express the area of triangle $\triangle ABC$ using $X$, follow these steps:

• Identify the base $BC = 6$.
• Identify the height as $AD = X$.
• Use the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
• Substitute the known values: $\text{Area} = \frac{1}{2} \times 6 \times X$.
• Simplify the expression: $\text{Area} = \frac{6X}{2}$.
• Further simplify: $\text{Area} = 3X$.

Comparing this with the choices given, choices B ($\frac{6X}{2}$) and C ($3X$) are both valid representations of the area.

Therefore, the correct answer is that choices B and C are correct.

Answers B and C are correct.

To solve this problem, we need to determine $X$ using the triangle area formula. Let's break it down step-by-step:

• Step 1: The area of a triangle $\Delta ABC$ is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
• Step 2: We substitute the given values where the base is $(X + 6)$ and the height is $5$: $22.5 = \frac{1}{2} \times (X + 6) \times 5$
• Step 3: Simplify the equation: $22.5 = \frac{1}{2} \times 5 \times (X + 6)$
• Step 4: Multiply out the constants: $22.5 = \frac{5}{2} \times (X + 6)$
• Step 5: Clear the fraction by multiplying both sides by 2: $45 = 5 \times (X + 6)$
• Step 6: Divide both sides by 5: $9 = X + 6$
• Step 7: Solve for $X$: $X = 9 - 6 = 3$

Therefore, the solution to the problem is $X = 3$.

To solve this problem, we will calculate the length of side $FE$ using the area formula for a triangle:

• Step 1: Use the formula for the area of a triangle: $A = \frac{1}{2} \times \text{base} \times \text{height}$.

• Step 2: Substitute the given values into the formula.
  We know that $A = 24 \, \text{cm}^2$ and $\text{height} = 8 \, \text{cm}$.

• Step 3: Set up the equation: $24 = \frac{1}{2} \times \text{base} \times 8$.

• Step 4: Simplify and solve for the base:$24 = \frac{8}{2} \times \text{base} \rightarrow 24 = 4 \times \text{base}$.

• Step 5: Solve for $\text{base}$: $\text{base} = \frac{24}{4} = 6 \, \text{cm}$.

Therefore, the side $FE$ of the triangle is 6 cm.

The area of triangle ABC is equal to:

$\frac{AD\times BC}{2}=2x+16$

As we are given the area of the triangle, we can insert this data into BC in the formula:

$\frac{AD\times(BD+DC)}{2}=2x+16$

$\frac{AD\times(x+5+3)}{2}=2x+16$

$\frac{AD\times(x+8)}{2}=2x+16$

We then multiply by 2 to eliminate the denominator:

$AD\times(x+8)=4x+32$

Divide by: $(x+8)$

$AD=\frac{4x+32}{(x+8)}$

We rewrite the numerator of the fraction:

$AD=\frac{4(x+8)}{(x+8)}$

We simplify to X + 8 and obtain the following:

$AD=4$

We now focus on triangle ADC and by use of the Pythagorean theorem we should find X:

$AD^2+DC^2=AC^2$

Inserting the existing data:

$4^2+(x+5)^2=(\sqrt{65})^2$

$16+(x+5)^2\text{ }=65/-16$

$(x+5)^2=49/\sqrt{}$

$x+5=\sqrt{49}$

$x+5=7$

$x=7-5=2$

The area of triangle ABC is:

$\frac{AB\times AC}{2}=S$

Into this formula, we insert the given data:

$\frac{AB\times(x+4)}{2}=4x+16$

$\frac{AB\times(x+4)}{2}=4(x+4)$

Notice that X plus 4 on both sides is reduced, and we are left with the equation:

$\frac{AB}{2}=4$

We then multiply by 2 and obtain the following:

$AB=4\times2=8$

If we now observe the triangle ABC we are able to find side BC using the Pythagorean Theorem:

$AB^2+AC^2=BC^2$

We first insert the existing data into the formula:

$8^2+(x+4)^2=BC^2$

We extract the root:

$BC=\sqrt{64+x^2+2\times4\times x+4^2}=\sqrt{x^2+8x+64+8}=\sqrt{x^2+8x+72}$

We can now calculate AD by using the formula to calculate the area of triangle ABC:

$S_{\text{ABC}}=\frac{AD\times BC}{2}$

We then insert the data:

$4x+16=\frac{AD\times\sqrt{x^2+8x+80}}{2}$

$AD=\frac{(4x+16)\times2}{\sqrt{x^2+8x+80}}=\frac{8x+32}{\sqrt{x^2+8x+80}}$

Let's calculate the area of triangle ABC:

$8x=\frac{(x+4)x}{2}$

Multiply by 2:

$16x=(x+4)x$

Divide by x:

$16=x+4$

Let's move 4 to the left side and change the sign accordingly:

$16-4=x$

$12=x$

Now let's calculate the area of the rectangle, multiply the length and width where BC equals 12 and AB equals 16:

$16\times12=192$