Question

В треугольнике DBC угол DBC = 90 град, угол BDC = 60 град, BD = 4 см. а) Между каким целыми числами заключена длина отрезка ВС? б) Найдите длину медианы PD.

Answer 1

Решение:

а) BC=BD*tg60=4*√3=6,93 см; 6 < 6,93 < 7  

б) длина медианы PD= √ [ (BC/2)^2 +BD^2 ] = √ [ (4√3/2)^2 +4^2 ] = 4√ [ (√3/2)^2 +1 ] = 2√7 см

Answer 2

а) BC = √ ( CD^2 - BD^2 ) = √ ( (BD/cosBDC)^2 - BD^2 ) =

= BD √ ( 1/(cosBDC)^2 - 1 ) = 4√ ( 1/(cos60)^2 - 1 ) =

= 4√3 см

6 < BC < 7  

б) длина медианы PD= √ [ (BC/2)^2 +BD^2 ] = √ [ (4√3/2)^2 +4^2 ] = 4√ [ (√3/2)^2 +1 ] = 2√7 см

Answer 3

Triangle DBC

In triangle DBC, we are given the following information: - Angle DBC = 90 degrees - Angle BDC = 60 degrees - BD = 4 cm

Let's solve the given questions step by step.

a) Between which whole numbers is the length of segment BC? To find the length of segment BC, we can use the trigonometric ratios in a right triangle. Since angle DBC is 90 degrees, triangle DBC is a right triangle.

Let's use the sine function to find the length of BC. The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse.

In triangle DBC, the side opposite angle BDC is BC, and the hypotenuse is BD. Therefore, we have:

sin(BDC) = BC/BD

We know that angle BDC is 60 degrees and BD is 4 cm. Substituting these values into the equation, we can solve for BC.

sin(60) = BC/4

Using the value of sin(60) (which is √3/2), we can solve for BC:

√3/2 = BC/4

Cross-multiplying, we get:

2 * BC = 4 * √3

Simplifying further:

BC = 2 * √3

Therefore, the length of segment BC is 2√3 cm.

b) Find the length of median PD. To find the length of median PD, we can use the properties of medians in a triangle. In a triangle, a median is a line segment that connects a vertex to the midpoint of the opposite side.

In triangle DBC, let's find the midpoint of side BC and call it M. Since BC has length 2√3 cm (as we found in part a), the midpoint M divides BC into two equal segments, each with length √3 cm.

Now, let's draw median PD. Median PD connects vertex D to the midpoint M of side BC. Since M is the midpoint of BC, DM is equal to BM, which is √3 cm.

Therefore, the length of median PD is equal to the length of DM, which is √3 cm.

So, to summarize: a) The length of segment BC is 2√3 cm. b) The length of median PD is √3 cm.

Please let me know if you need any further clarification or assistance!