You have two pieces of wood that will make up two sides of a triangular picture frame. One is 8 in. long and the other is 11 in. long. What is the range of possible lengths for the third side of the frame? Please show all work in order to get full credit for this problem. (4 points)
In a triangle, the length of any one side must be less than the sum of the lengths of the other two sides.
Let's denote the length of the third side as x.
According to the given condition, the length of the third side must be less than the sum of the lengths of the other two sides. The sum of the lengths of the other two sides is 8 + 11 = 19.
Therefore, x < 19.
However, the length of the third side must also be greater than the difference between the lengths of the other two sides. The difference between the lengths of the other two sides is |8 - 11| = 3.
Therefore, x > 3.
Combining these two conditions, we have 3 < x < 19.
So, the range of possible lengths for the third side of the frame is 3 to 19 inches.
In order to find the length of BC¯¯¯¯¯¯¯¯, we need to use the concept of the midpoint and medians in a triangle.
A median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. Therefore, DB¯¯¯¯¯¯¯¯ is the median of triangle ADC since it connects vertex D to the midpoint of side AC¯¯¯¯¯¯¯¯.
Since DB¯¯¯¯¯¯¯¯ is the median, it divides AC¯¯¯¯¯¯¯¯ into two equal segments. Let's call the length of each segment x. So, AC = x + x = 2x.
Given that AC = 70, we can set up the equation 2x = 70.
Solving for x, we divide both sides by 2: x = 70/2 = 35.
Therefore, each segment of AC¯¯¯¯¯¯¯¯ is 35 units long.
Now, we can find the length of BC¯¯¯¯¯¯¯¯. Since B is the midpoint of AC¯¯¯¯¯¯¯¯, BC¯¯¯¯¯¯¯¯ is also equal to x, which is 35 units.
So, the length of BC¯¯¯¯¯¯¯¯ is 35 units.
To find the possible values of x, we need to use the angle-angle similarity postulate. This postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
In this case, angle ABD in triangle ABD is congruent to angle CDB in triangle CDB.
Since angle B in triangle ABD is given as 63 degrees, and angle B in triangle CDB is (8x + 7) degrees, we can set up an equation:
63 = 8x + 7
Solving for x, we subtract 7 from both sides:
63 - 7 = 8x
56 = 8x
Dividing both sides by 8:
x = 56/8
x = 7
Therefore, the only possible value of x is 7.
In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Based on this concept, we can write an inequality to represent the possible lengths for the third side, x.
The inequality can be written as:
13 + 19 > x
This can be simplified to:
32 > x
So, the inequality that represents the possible lengths for the third side, x, is:
x < 32
Find the length of BC¯¯¯¯¯¯¯¯ . AC = 70 and DB¯¯¯¯¯¯¯¯ is the median of the triangle. Please show all work in order to get credit for this problem
there is an image of a triangle, one angle is labeled D, another labeled A, another labeled C, and the point at the end of the median line is labeled B
What is the possible values of x? The diagram is not to scale. Please show all steps in order to earn full credit.
there in an image showing two triangles connected to each other, the angles of triangle 1 are labeled ABD, and triangle 2 is CDB. angle B, on the side of triangle 1, is 63 degrees. line AD equals 11 units in length. line DC equals 8 units in length. the angle B, on the side of triangle 2, is (8x+7) degrees.