How to find the area of a triangle - ISEE Middle Level Quantitative Reasoning

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Question

A triangle has base 80 inches and area 4,200 square inches. What is its height?

Answer

Use the area formula for a triangle, setting :

$A = \frac{1}{2} bh$

$4200 = \frac{1}{2}\cdot 80 \cdot h = 40h$

$h = 4200 \div 40 = 105$ inches

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Question

The sum of the lengths of the legs of an isosceles right triangle is one meter. What is its area in square centimeters?

Answer

The legs of an isosceles right triangle have equal length, so, if the sum of their lengths is one meter, which is equal to 100 centimeters, each leg measures half of this, or

$100 \div 2 = 50$ centimeters.

The area of a triangle is half the product of its height and base; for a right triangle, the legs serve as height and base, so the area of the triangle is

$\frac{1}{2} \times 50 \times 50 = 1,250$ square centimeters.

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Question

Figure NOT drawn to scale

Square has area 1,600. ; . Which of the following is the greater quantity?

(a) The area of $\bigtriangleup AZX$

(b) The area of $\bigtriangleup BZY$

Answer

Square has area 1,600, so the length of each side is $\sqrt{1,600 }= 40$ .

Since ,

Therefore, .

$\bigtriangleup AZX$ has as its area $\frac{1}{2} \cdot AX \cdot AZ$ ; $\bigtriangleup BZY$ has as its area $\frac{1}{2} \cdot BX \cdot BZ$ .

Since and , it follows that

$BX \cdot BZ > AX \cdot AZ$

and

$\frac{1}{2} \cdot BX \cdot BZ >\frac{1}{2} \cdot AX \cdot AZ$

$\bigtriangleup BZY$ has greater area than $\bigtriangleup AZX$ .

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Question

The above figure depicts Square . , , and are the midpoints of $\overline{AD}$ , $\overline{BC}$ , and $\overline{AB}$ , respectively.

$\bigtriangleup AZX$ has area . What is the area of Square ?

Answer

Since , , and are the midpoints of $\overline{AD}$ , $\overline{BC}$ , and $\overline{AB}$ , if we call the length of each side of the square, then

$AX = AZ = \frac{1}{2}s$

The area of $\bigtriangleup AZX$ is half the product of the lengths of its legs:

$\frac{1}{2} \cdot AX \cdot AZ$

$= \frac{1}{2} \cdot \frac{1}{2}s \cdot \frac{1}{2}s$

$= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\cdot s \cdot s$

$= \frac{1 \cdot 1 \cdot 1}{2\cdot 2 \cdot 2}\cdot s ^{2}$

$= \frac{1}{8} s ^{2}$

The area of the square is the square of the length of a side, which is $s^{2}$ . This is eight times the area of $\bigtriangleup AZX$ , so the correct choice is

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Question

Which of the following is the greater quantity?

(a) The area of the above triangle

(b) 800

Answer

The area of a right triangle is half the product of the lengths of its legs, which here are 25 and 60. So

$A = \frac{1}{2} \cdot 25 \cdot 60 = 750$

which is less than 800.

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Question

The above figure gives the lengths of the three sides of the triangle in feet. Give its area in square inches.

Answer

The area of a right triangle is half the product of the lengths of its legs, which here are feet and feet.

Multiply each length by 12 to convert to inches - the lengths become and . The area in square inches is therefore

$\frac{1}{2} (12A )(12 B) = \frac{1}{2} \cdot 12 \cdot 12 \cdot A \cdot B = 72AB$ square inches.

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Question

Refer to the above figure. Which is the greater quantity?

(a) The perimeter of the triangle

(b) 3 feet

Answer

The perimeter of the triangle - the sum of the lengths of its sides - is

inches.

3 feet are equivalent to $3 \times 12 = 36$ inches, so this is the greater quantity.

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Question

Figure NOT drawn to scale.

In the above diagram, Square has area 400. Which is the greater quantity?

(a) The area of $\bigtriangleup SQE$

(b) The area of $\bigtriangleup UAR$

Answer

Square has area 400, so its common sidelength is the square root of 400, or 20. Therefore,

.

The area of a right triangle is half the product of the lengths of its legs.

$\bigtriangleup SQE$ has legs $\overline{SQ}$ and $\overline{SE}$ , so its area is

$\frac{1}{2} \cdot SQ \cdot SE = \frac{1}{2} \cdot 9 \cdot 20 = 90$ .

$\bigtriangleup UAR$ has legs $\overline{UA}$ and $\overline{AR}$ , so its area is

$\frac{1}{2} \cdot AR \cdot UA = \frac{1}{2} \cdot 11 \cdot 20 = 110$ .

$\bigtriangleup UAR$ has the greater area.

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Question

Figure NOT drawn to scale

The above diagram depicts Parallelogram . Which is the greater quantity?

(a) The area of $\bigtriangleup AMD$

(b) The area of $\bigtriangleup BMC$

Answer

Opposite sides of a parallelogram have the same measure, so

Base $\overline{AM}$ of $\bigtriangleup AMD$ and base $\overline{MB}$ of $\bigtriangleup BMC$ have the same length; also, as can be seen below, both have the same height, which is the height of the parallelogram.

Therefore, the areas of $\bigtriangleup AMD$ and $\bigtriangleup BMC$ have the same area - $\frac{1}{2} \cdot 60 \cdot h = 30h$ .

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Question

What is the area of a triangle with a base of and a height of ?

Answer

The formula for the area of a triangle is $\dpi{100} Area=\frac{1}{2}\times base\times height$ .

Plug the given values into the formula to solve:

$\dpi{100} Area=\frac{1}{2}\times 12\times 3$

$\dpi{100} Area=\frac{1}{2}\times 36$

$\dpi{100} Area=18$

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Question

A triangle has a base of $\small 43$ and an area of $\small 2042.5$ . What is the height?

Answer

The area of a triangle is found by multiplying the base by the height and dividing by two:

$\small \frac{b\cdot h}{2}$

In this problem we are given the base, which is $\small 43$ , and the area, which is $\small 2042.5$ . First we write an equation using $\small h$ as our variable.

$\small \frac{43\cdot h}{2}=2042.5$

To solve this equation, first multply both sides by $\small 2$ , becuase multiplication is the opposite of division and therefore allows us to eliminate the $\small 2$ .

The left-hand side simplifies to:

$\small \frac{43\cdot h}{2}\cdot 2=43\cdot h$

The right-hand side simplifies to:

$\small 2042.5\cdot 2=4085$

So our equation is now:

$\small \small 43\cdot h=4085$

Next we divide both sides by $\small \small 43$ , because division is the opposite of multiplication, so it allows us to isolate the variable by eliminating $\small \small 43$ .

$\small \frac{43\cdot h}{43}= h$

$\small \frac{4085}{43}=95$

$\small h=95$

So the height of the triangle is $\small 95$ .

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Question

Bill paints a triangle on his wall that has a base parallel to the ground that runs from one end of the wall to the other. If the base of the wall is 8 feet, and the triangle covers 40 square feet of wall, what is the height of the triangle?

Answer

In order to find the area of a triangle, we multiply the base by the height, and then divide by 2.

$\small \frac{b\cdot h}{2}$

In this problem we are given the base and the area, which allows us to write an equation using $\small h$ as our variable.

$\small \frac{8\cdot h}{2}=40$

Multiply both sides by two, which allows us to eliminate the two from the left side of our fraction.

The left-hand side simplifies to:

$\small \frac{8\cdot h}{2}\cdot 2=8\cdot h$

The right-hand side simplifies to:

$\small 40\cdot 2=80$

Now our equation can be rewritten as:

$\small 8\cdot h=80$

Next we divide by 8 on both sides to isolate the variable:

$\small \frac{8\cdot h}{8}=h$

$\small \frac{80}{8}=10$

$\small h=10$

Therefore, the height of the triangle is $\small 10 : feet$ .

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Question

Note: Figure NOT drawn to scale.

The above triangle has area 36 square inches. If $x = 4.5 \textrm{ in}$ , then what is ?

Answer

The area of a triangle is one half the product of its base and its height - in the above diagram, that means

$A = \frac{1}{2}xy$ .

Substitute , and solve for .

$\frac{1}{2} \cdot 4.5 \cdot y = 36$

$2.25 y \div 2.25= 36\div 2.25$

$y = 16 \textrm{ in}$

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Question

Please use the following shape for the question.

What is the area of this shape?

Answer

From this shape we are able to see that we have a square and a triangle, so lets split it into the two shapes to solve the problem. We know we have a square based on the 90 degree angles placed in the four corners of our quadrilateral.

Since we know the first part of our shape is a square, to find the area of the square we just need to take the length and multiply it by the width. Squares have equilateral sides so we just take 5 times 5, which gives us 25 inches squared.

We now know the area of the square portion of our shape. Next we need to find the area of our right triangle. Since we know that the shape below the triangle is square, we are able to know the base of the triangle as being 5 inches, because that base is a part of the square's side.

To find the area of the triangle we must take the base, which in this case is 5 inches, and multipy it by the height, then divide by 2. The height is 3 inches, so 5 times 3 is 15. Then, 15 divided by 2 is 7.5.

We now know both the area of the square and the triangle portions of our shape. The square is 25 inches squared and the triangle is 7.5 inches squared. All that is remaining is to added the areas to find the total area. Doing this gives us 32.5 inches squared.

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Question

The hypotenuse of a right triangle is 25 inches; it has one leg 15 inches long. Give its area in square feet.

Answer

The area of a right triangle is half the product of the lengths of its legs, so we need to use the Pythagorean Theorem to find the length of the other leg. Set :

$a^{2} = c^{2} - b^{2}$

$a^{2} = 25^{2} - 15^{2}$

$a^{2} = 625-225$

$a^{2} =400$

$a = \sqrt{400} = 20$

The legs are 15 and 20 inches long. Divide both dimensions by 12 to convert from inches to feet:

$20 \div 12 = \frac{5}{3}$ feet

$15 \div 12 = \frac{5}{4}$ feet

Now find half their product:

$A = \frac{1}{2} \times \frac{5}{3}\times \frac{5}{4} = \frac{25}{24} = 1 \frac{1}{24}$ square feet

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Question

The hypotenuse of a right triangle is feet; it has one leg feet long. Give its area in square inches.

Answer

The area of a right triangle is half the product of the lengths of its legs, so we need to use the Pythagorean Theorem to find the length of the other leg. Set :

$a^{2} = c^{2} - b^{2}$

$a^{2} = 39 ^{2} - 36^{2}$

$a^{2} = 1,521 - 1,296$

$a^{2} = 225$

$a = \sqrt{225} = 15$

The legs have length and feet; multiply both dimensions by to convert to inches:

$15 \times 12 = 180$ inches

$36 \times 12 = 432$ inches.

Now find half the product:

$A = \frac{1}{2} \cdot 180 \cdot 432 = 38,880\ in^{2}$

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Question

What is the area of the triangle?

Answer

Area of a triangle can be determined using the equation:

$\small A=\frac{1}{2}bh$

$\small A=\frac{1}{2}(14)(5)=35$

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Question

The three angles of a triangle are labeled , , and . If is $97^{\circ}$ , what is the value of ?

Answer

Given that the three angles of a triangle always add up to 180 degrees, the following equation can be used:

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Question

A triangle has a height of 9 inches and a base that is one third as long as the height. What is the area of the triangle, in square inches?

Answer

The area of a triangle is found by multiplying the base times the height, divided by 2.

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

Given that the height is 9 inches, and the base is one third of the height, the base will be 3 inches.

$b=h\div3=9\div3=3$

We now have both the base (3) and height (9) of the triangle. We can use the equation to solve for the area.

$\text{Area} =\frac{9\times 3}{2}$

$\text{Area} =\frac{27}{2}$

The fraction cannot be simplified.

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Question

What is the area (in square feet) of a triangle with a base of feet and a height of feet?

Answer

The area of a triangle is found by multiplying the base times the height, divided by .

$Area =\frac{base\cdot height}{2}$

$Area =\frac{4\cdot 7}{2}$

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