Question

KLNP is a square with a perimeter of 128.
Column A: MQ
Column B: 42

• A. Quantity A is greater
• B. Quantity B is greater
• C. The two quantities are equal
• D. The relationship cannot be determined from the information given

Hint:

Recognize common Pythagorean triples to save time. In this problem, the triangle with sides 16, MT, and 20 is a multiple of the basic 3-4-5 triple. \(16 = 4 \times 4\), \(20 = 5 \times 4\), so the missing side must be \(3 \times 4 = 12\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem combines properties of a square with the Pythagorean theorem in an isosceles triangle. We need to calculate a total length which is composed of the side of the square and the altitude of the triangle.
Step 2: Key Formula or Approach:
1. Perimeter of a square = \(4 \times \text{side}\).
2. Pythagorean theorem: In a right-angled triangle, \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse.
Step 3: Detailed Explanation:
Find the side length of the square KLNP.
The perimeter is given as 128.
\[ \text{Side length} = \frac{\text{Perimeter}}{4} = \frac{128}{4} = 32 \] So, \(KL = LN = NP = PK = 32\).
Find the height of the triangle LMN.
Triangle LMN is an isosceles triangle since \(LM = MN = 20\). The base of this triangle is the side LN of the square, so \(LN = 32\).
Let's draw an altitude from vertex M to the base LN. Let the point of intersection be T. In an isosceles triangle, the altitude to the base is also the median, so it bisects the base.
\[ LT = TN = \frac{LN}{2} = \frac{32}{2} = 16 \] Now we have a right-angled triangle, \(\triangle MTL\), with:
- Hypotenuse \(LM = 20\)
- Leg \(LT = 16\)
- Leg \(MT\) (the height of the triangle)
Using the Pythagorean theorem:
\[ MT^2 + LT^2 = LM^2 \] \[ MT^2 + 16^2 = 20^2 \] \[ MT^2 + 256 = 400 \] \[ MT^2 = 400 - 256 = 144 \] \[ MT = \sqrt{144} = 12 \] The height of the triangle LMN is 12.
Calculate the total length MQ.
The diagram shows that the total length MQ is the sum of the height of the triangle (MT) and the side length of the square (which is the vertical distance from the line LN to the line KP, equal to LK or NP).
\[ MQ = MT + \text{side of square} \] \[ MQ = 12 + 32 = 44 \] Step 4: Comparing the Quantities:
Column A: \(MQ = 44\)
Column B: 42
Since \(44 \textgreater 42\), the quantity in Column A is greater.