Question

MNK — равнобедренный треугольник с основанием NM. P_{MNK}=500 мм. Чему равна биссектриса KR если периметр треугольника KRN составляет 43 см? Вырази ответ в см.

Answer 1

Ответ:

Так как  ΔMNK - равнобедренный, то MK=NK . Пусть  NK=MK=a .

И биссектриса KR является высотой . Пуcть  KR=h .

Также MR=NR . Пусть  MR=b .

Периметр P(MNK)=500 мм=50 см .  Периметр Р(KRN)=43 см .

P(MNK)=MN+MK+NK=2b+2a=50 см  ,   2(a+b)=50  ,  a+b=25 см

P(KRN)=NK+NR+KR=a+b+h=43 см  ,  (a+b)+h=43  ,  25+h=43  ,  h=43-25  ,

             h=18 см

           KR=18 cм

Answer 2

Calculation of the Bisector KR of Triangle KRN

To find the length of the bisector KR of triangle KRN, we need to use the given information about the triangle and apply the properties of a triangle's bisector.

Given: - Triangle MNK is an isosceles triangle with base NM. - The length of the perpendicular from the vertex M to the base NM is 500 mm. - The perimeter of triangle KRN is 43 cm.

To find the length of the bisector KR, we can use the following steps:

Step 1: Convert the given measurements to a consistent unit. - Since the length of the perpendicular is given in millimeters (mm), we need to convert the perimeter from centimeters (cm) to millimeters (mm) for consistency. - 1 cm = 10 mm, so the perimeter of triangle KRN is 43 cm * 10 mm/cm = 430 mm.

Step 2: Apply the property of the bisector in an isosceles triangle. - In an isosceles triangle, the bisector from the vertex angle divides the base into two equal segments. - Therefore, the length of the segment NR is equal to the length of the segment KR.

Step 3: Calculate the length of the segment NR. - The perimeter of triangle KRN is the sum of the lengths of its sides: KR + RN + NK = 430 mm. - Since KR = NR, we can rewrite the equation as: KR + KR + NK = 430 mm. - Simplifying the equation, we get: 2KR + NK = 430 mm.

Step 4: Calculate the length of the bisector KR. - We know that the length of the perpendicular from the vertex M to the base NM is 500 mm. - In an isosceles triangle, the perpendicular from the vertex angle to the base bisects the base. - Therefore, the length of the segment NK is equal to half the length of the perpendicular, which is 500 mm / 2 = 250 mm.

Step 5: Substitute the values into the equation and solve for KR. - 2KR + 250 mm = 430 mm. - Subtracting 250 mm from both sides, we get: 2KR = 430 mm - 250 mm = 180 mm. - Dividing both sides by 2, we find: KR = 180 mm / 2 = 90 mm.

Answer:

The length of the bisector KR of triangle KRN is 90 cm.