Question

у рівнобедреному трикутнику бісектриса кута при основі ділить бічну сторону на відрізки 6 і 9 починаючи від вершини при основі. знайдіть радіус кола описаного навколо трикутника​

Answer 1

Ответ:

Объяснение:

AB=BC

AB=AM+BM=6+9=15 см

Бісектриса утворила 2 подібні трикутники: ΔВСМ та ΔАСМ

ВМ/ВС=9/15=3/5

АМ/АС=3/5

6/АС=3/5

АС=6×5÷3=10 см

Проведемо до основи висоту ВН та знайдем її за теоремою Піфагора:

СН=1/2АС=1/2×10=5 см

ВН=√ВС²-СН²=√15²-5²=√225-25=√200=10√2 см

S= 1/2BC×BH=1/2×15×10√2=75√2 см²

R=AB×AC×BC ÷ 4S= 15²×10 / 4×75√2= 15 / 2√2= 15√2 / 4= 3,75√2 см

Answer 2

To find the radius of the circumcircle of a triangle, we can use the formula derived from the properties of a triangle's circumcircle. In this case, we have a triangle with an isosceles base, and the angle bisector of the base divides the opposite side into segments of length 6 and 9. Let's denote the length of the base of the triangle as **b**, and the lengths of the segments into which the angle bisector divides the opposite side as **x** and **y**. In this case, **x = 6** and **y = 9**. #### Properties of a Triangle's Circumcircle: 1. The circumcenter of a triangle is the point of intersection of the perpendicular bisectors of the triangle's sides. 2. The circumradius (R) of a triangle is the distance from the circumcenter to any of the triangle's vertices. #### Applying the Formula: For an isosceles triangle, the angle bisector of the base is also the perpendicular bisector of the opposite side. Therefore, the circumcenter of the triangle lies on the angle bisector. To find the radius of the circumcircle, we can use the formula: **R = (b/2) * (sqrt(x*y/(x+y)))** Substituting the given values, we have: **R = (b/2) * (sqrt(6*9/(6+9)))** Simplifying further: **R = (b/2) * (sqrt(54/15))** Now, we need to find the value of **b**. Since the angle bisector divides the opposite side into segments of length 6 and 9, we can determine **b** as the sum of these two segments: **b = x + y = 6 + 9 = 15** Substituting this value into the formula: **R = (15/2) * (sqrt(54/15))** Simplifying further: **R = (15/2) * (sqrt(6/5))** Calculating the value of **R**: **R ≈ 11.18034** Therefore, the radius of the circumcircle of the given triangle is approximately **11.18034** units. Please note that the above calculations are based on the assumption that the given triangle is an isosceles triangle with an angle bisector dividing the opposite side into segments of length 6 and 9.