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Find the value of x in a right isosceles triangle with hypotenuse 8 (round to nearest tenth)
Mathematics
Grade 8
Question Content
Determine the value of x. Round decimals to the nearest tenth. The figure is a right isosceles triangle with legs of length x and hypotenuse of length 8.
Correct Answer
5.7
Detailed Solution Steps
1
Step 1: Identify the type of triangle and relevant theorem. This is a right isosceles triangle, so we use the Pythagorean Theorem, which states that for a right triangle with legs a, b and hypotenuse c: $a^2 + b^2 = c^2$.
2
Step 2: Substitute the known values into the theorem. Here, a = x, b = x, c = 8, so the equation becomes $x^2 + x^2 = 8^2$.
3
Step 3: Simplify the equation. Combine like terms on the left side: $2x^2 = 64$.
4
Step 4: Solve for $x^2$ by dividing both sides of the equation by 2: $x^2 = 32$.
5
Step 5: Solve for x by taking the square root of both sides: $x = \\sqrt{32}$. Calculate the decimal value: $\\sqrt{32} \\approx 5.656$.
6
Step 6: Round the result to the nearest tenth: $x \\approx 5.7$.
Knowledge Points Involved
1
Pythagorean Theorem
The theorem applies to right-angled triangles, stating that the sum of the squares of the lengths of the two legs (the sides forming the right angle) is equal to the square of the length of the hypotenuse (the side opposite the right angle). The formula is $a^2 + b^2 = c^2$, where a and b are legs, c is the hypotenuse. It is used to find unknown side lengths in right triangles.
2
Right Isosceles Triangle
A right triangle where the two legs (the sides adjacent to the right angle) are equal in length. This means the two non-right angles are also equal, each measuring 45 degrees. When solving for side lengths, the two equal legs can be substituted with the same variable in the Pythagorean Theorem.
3
Rounding Decimals to the Nearest Tenth
A rounding rule where you look at the hundredths place (the second digit after the decimal point) of a number. If this digit is 5 or greater, you round up the tenths place digit; if it is less than 5, you keep the tenths place digit the same. This is used to simplify decimal numbers to a specified precision.
4
Square Root Calculation
The square root of a number is a value that, when multiplied by itself, gives the original number. For non-perfect squares, we calculate an approximate decimal value using a calculator or estimation methods. It is used to solve equations where a variable is squared, such as in the Pythagorean Theorem.
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