Questions: На рис. 38 K L M N- прямоугольник, угол C=90°, AC=BC, KL:LM=2:5, PKLMN=56 см. Найдите AB.

Transcript text: На рис. $38 K L M N-$ прямоугольник, $\angle C=90^{\circ}, A C=B C$, $K L: L M=2: 5, P_{K L M N}=56 \mathrm{~cm}$. Найдите $A B$.

Solution

Solution Steps

Step 1: Understand the Problem

We are given a right triangle $ \triangle KLMN $ with $ \angle C = 90^\circ $, $ AC = BC $, $ KL:LM = 2:5 $, and the perimeter $ P_{KLMN} = 56 $ cm. We need to find the length of $ AB $.

Step 2: Identify Key Relationships

Since $ \angle C = 90^\circ $ and $ AC = BC $, $ \triangle ABC $ is an isosceles right triangle. This implies $ AC = BC = x $ and $ AB = x\sqrt{2} $.

Step 3: Use Given Ratios

Given $ KL:LM = 2:5 $, let $ KL = 2k $ and $ LM = 5k $. The perimeter of $ \triangle KLMN $ is the sum of all sides: \[ KL + LM + MN + NK = 56 \]

Step 4: Calculate the Perimeter

Since $ \triangle KLMN $ is a right triangle, we can use the Pythagorean theorem: \[ KL^2 + LM^2 = KN^2 \] \[ (2k)^2 + (5k)^2 = KN^2 \] \[ 4k^2 + 25k^2 = KN^2 \] \[ 29k^2 = KN^2 \] \[ KN = \sqrt{29}k \]

Step 5: Solve for $ k $

The perimeter equation becomes: \[ 2k + 5k + \sqrt{29}k + k = 56 \] \[ 8k + \sqrt{29}k = 56 \] \[ k(8 + \sqrt{29}) = 56 \] \[ k = \frac{56}{8 + \sqrt{29}} \]

Step 6: Simplify $ k $

To simplify $ k $, rationalize the denominator: \[ k = \frac{56(8 - \sqrt{29})}{(8 + \sqrt{29})(8 - \sqrt{29})} \] \[ k = \frac{56(8 - \sqrt{29})}{64 - 29} \] \[ k = \frac{56(8 - \sqrt{29})}{35} \] \[ k = \frac{56}{35}(8 - \sqrt{29}) \] \[ k = \frac{8}{5}(8 - \sqrt{29}) \]

Step 7: Calculate $ AB $

Since $ AC = BC = x $ and $ AB = x\sqrt{2} $, we need to find $ x $. Given $ AC = BC $ and the perimeter of $ \triangle ABC $ is $ 56 $ cm, we can use the relationship: \[ AC + BC + AB = 56 \] \[ x + x + x\sqrt{2} = 56 \] \[ 2x + x\sqrt{2} = 56 \] \[ x(2 + \sqrt{2}) = 56 \] \[ x = \frac{56}{2 + \sqrt{2}} \]

Step 8: Simplify $ x $

Rationalize the denominator: \[ x = \frac{56(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})} \] \[ x = \frac{56(2 - \sqrt{2})}{4 - 2} \] \[ x = \frac{56(2 - \sqrt{2})}{2} \] \[ x = 28(2 - \sqrt{2}) \]