An isosceles right triangle has an area of 98cm squared. Find the length of each leg. The formula of finding the area of a triangle with one right angle is: (b x h)/2 You know that an isoceles triangles has two equal sides. Those equal sides are the base and height. This means that both the base height with have the same value. After you get the answer of the two isoceles legs, you need to figure out the other leg. To solve this by using the formula: a^2+b^2=c^2 a would be on leg length b would be the other leg length You just fill in the values and you get your answer. =) this did not help If the area is 98 then the sides would be equal to b x h / 2. So they would be 98=b x h /2 98 x 2= b x h 196= b x h (we know that the base and the height are the same so you can just find the square root of 196) square root of 196= 14×14 So the two sides are 14 and 14 Now we need to find the third side. We will use the formula: a^2+b^2=c^2 14^2+14^2=c^2 196+196=c^2 329=c^2 So the side lengths of the triangle are (14,14, square root of 329) f
The Correct Answer and Explanation is :
To find the side lengths of the isosceles right triangle with an area of 98 cm², let’s go through the correct steps systematically.
Step 1: Identify Given Information
We know that the area of the triangle is 98 cm². The formula for the area of a right triangle is:
[
\frac{b \times h}{2} = 98
]
Since the triangle is isosceles, the base ((b)) and height ((h)) are equal, meaning:
[
\frac{x \times x}{2} = 98
]
[
\frac{x^2}{2} = 98
]
Step 2: Solve for (x) (Leg Length)
Multiply both sides by 2 to eliminate the fraction:
[
x^2 = 196
]
Take the square root:
[
x = \sqrt{196} = 14
]
Thus, both legs of the triangle are 14 cm.
Step 3: Find the Hypotenuse ((c))
We now use the Pythagorean theorem:
[
a^2 + b^2 = c^2
]
Substituting (a = 14) and (b = 14):
[
14^2 + 14^2 = c^2
]
[
196 + 196 = c^2
]
[
392 = c^2
]
Taking the square root:
[
c = \sqrt{392}
]
Since (\sqrt{392} = 2\sqrt{98}), we leave it in simplified radical form.
Final Answer
The side lengths of the isosceles right triangle are:
[
(14, 14, \sqrt{392} \text{ or } 2\sqrt{98})
]
Explanation
- Understanding the formula: The area of a right triangle is found using (\frac{b \times h}{2}). Since the given triangle is isosceles, we set the base equal to the height.
- Solving for leg length: We used algebraic manipulation to solve for (x), yielding 14 cm for both legs.
- Applying the Pythagorean theorem: Since this is a right triangle, we used (a^2 + b^2 = c^2) to find the hypotenuse.
- Simplification: The hypotenuse simplifies to (2\sqrt{98}), which is the most simplified radical form.
Thus, the correct answer is (14 cm, 14 cm, 2√98 cm or √392 cm).