We read your photo as:

A regular tetrahedron is a three-dimensional shape with four faces, each of which is an equilateral triangle. All edges are of equal length, in this case, 4 cm.

Mathematics

ScanSolve AI Answer

Step-by-step solution

3 steps

To solve this problem, we need to visualize the tetrahedron's net and then find the shortest path on the unfolded surface.

A regular tetrahedron is a three-dimensional shape with four faces, each of which is an equilateral triangle. All edges are of equal length, in this case, 4 cm.

The path from P to S passes over the edge QR at its midpoint T. This means the path lies on two faces: △PQR\triangle PQR and △SQR\triangle SQR. To find the straight path, we need to unfold these two faces into a single plane.

  1. Draw the net of the relevant faces.

    1. Draw an equilateral triangle PQR with side length 4 cm.

    2. Draw another equilateral triangle SQR, sharing the edge QR, such that vertex S is on the opposite side of QR from vertex P. This forms a larger triangle (or a rhombus if P, Q, S, R were coplanar, but here P and S are vertices of two triangles sharing a base).

      The net will look like this:

            P
           / \
          /   \
         /     \
        Q-------R
         \     /
          \   /
           \ /
            S
      
  2. Locate the midpoint T and draw the path.

    1. Mark the midpoint T on the edge QR.
    2. The straight path from P to S, passing through T, is a straight line segment connecting P and S on this unfolded net.
  3. Calculate the length of the path PS.

    1. The line segment PT is the altitude (height) of the equilateral triangle PQR from vertex P to the base QR. The formula for the altitude hh of an equilateral triangle with side length aa is h=32ah = \frac{\sqrt{3}}{2}a. For △PQR\triangle PQR, a=4a = 4 cm. So, the length of PT=32×4=23PT = \frac{\sqrt{3}}{2} \times 4 = 2\sqrt{3} cm.

    2. Similarly, the line segment ST is the altitude of the equilateral triangle SQR from vertex S to the base QR. For △SQR\triangle SQR, a=4a = 4 cm. So, the length of ST=32×4=23ST = \frac{\sqrt{3}}{2} \times 4 = 2\sqrt{3} cm.

    3. On the unfolded net, P, T, and S are collinear. Therefore, the total length of the straight path PS is the sum of PT and ST. Length of path PS = PT+ST=23 cm+23 cm=43 cmPT + ST = 2\sqrt{3} \text{ cm} + 2\sqrt{3} \text{ cm} = 4\sqrt{3} \text{ cm}.

    4. To get a numerical value, we use 3≈1.732\sqrt{3} \approx 1.732. Length of path PS ≈4×1.732=6.928\approx 4 \times 1.732 = 6.928 cm.

    The length of the straight path of PS through the midpoint T over the edge QR is 43 cm\boxed{4\sqrt{3} \text{ cm}} or approximately 6.93 cm\boxed{6.93 \text{ cm}}.

Was this helpful?

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question